Random functions with given time correlation

INFORMATION

SCIENCES

11, 141-185 (1976)

141

Random Functions with Given Time Correlation JACQUES MAURIN 22 Rue Jean Richqin, 91120 Palaiseau, France Communicated

by Robert Fortet

ABSTRACT First, on any sequence of real numbers (q), AE[ - A, + A] cZ, the pseudo probability Pr(x,x’) of the event xx E[x,x’[ is defined to be the limit when Ada, of the ratio of the number of x, E [ x, x’[ to the total number of xx. The a.d.f. (asymptotic distribution function) of the sequence is then defined by F(x)= Pr(- co,x); it possesses the properties of a d.f. (distribution function). Consequently, what is said below applies equally to a sequence of r.v. (random variables) or to a sequence of p.r.v. (pseudorandom variables) consisting of a sequence(n~h),nE[-N,+N]~Zofsequences.xx,XE[-A,+A]~Z. A Weyl’s polynomial cp,Q is a polynomial such that one of its coefficieints other than q(O) is irrational. Then any sequence “xh = cp,(X), the fractional part of cp,(h), h E [ - A, + A] c Z, is asymptotically equidistributed on [0, lj? A property is given which permits the construction of a sequence (“xh), n E [ - N, + N] c Z of pseudostochastically independent sequences “xX, h E [ - A, + A] c Z. It is known that setting Y, = F(- “(X,,), it is possible to transform any sequence of r.v. X,

INTRODUCTION HISTORICAL

NOTE

The origin of the present paper is a problem of turbulence in fluids mechanics. The question was to mathematically define and calculate a threedimensional field of velocity fluctuations analogous to those given experimentally and having space-time correlations similar to those measured in a wind tunnel [lA]. For that purpose, refer the space to an orthonormal frame with unitary vectors l,, l,, 1,. Consider a permanent turbulent flow, having a mean speed V parallel to l,, assumed to be the speed unity. Define the velocity fluctuation u(x, t) = l,*ul(x, t) + 12.u2(x,t) + 13-z+(x,t) at a point M having a position OM = x = 1,-x, + 1,-x,+ 1,.x, and at the instant t, by the difference between the speed at that point and V. QAmerican Elsevier Publishing Company, Inc., 1976

142

JACQUES

MAURIN

or p.r.v. ,,xA, n E[ - N, + N]cZ, into a sequence having any sequence of d.f. or a.d.f. F,(x) previously given. Conditions are defined on such sequences (X,) of independent r.v. or p.r.v. that allow one to obtain a r.f. (random function) Y(r) having, to an approximation as good as required, a time correlation function g,(B) = lim L

~-rrnzT

function possesses a spectral density. The r.f. above Y(r) has a d.f. Q(y,r) S(y,y’;r,r’) of (Y(f), Y(r’)). If Q(y)=

lim L Tim 2Tl_TTQ(y,l)d

exist, then the a.d.f. Q(y)=

and

1’

-T

Y (1) Y (t + O)dt previously given, if the latter

and admits a j.d.f. (joint distribution

s(y)=

$rn, & I-‘,1 r(,).&r

function)

$im, &l/T’T S(uu;kl’)drdr -+ - T,T

[ 1ro)
the set of t for which Y(r)y],>f Y(r) is stochastically equal to its m.d.f. (mean stochastic distribution function Q(y); that property is an ergodic property.

Between two points M’,M”, the space-time tion 8 = t” - t’ is a tensor defined by

gV(x’,x”;B)=

lim -!T--.co~T

correlation

* ui(x’,t’).Ui(X”,“+ s _T

In the experimental flow studied, the longitudinal is spatially homogeneous and temporally stationary; only on space 4=x” -x’ and time 8 separations, representable by the pattern

for a time separa-

e)dt’.

space-time correlation g,, in other words it depends and it is approximately

which possesses the cylindrical symmetry, 1, being any direction perpendicular to ll. The autocorrelation function g(e)=g(O,e) of the speed u,(t)= u,(O,t) is an even function, maximal at the origin 8=0, and tending to 0 as 8+oc; this defines u,(t) as a pseudorandom function [3]. To simulate that at least autocorrelation

u,(t), start with a Weyl’s polynomial

[3,5] p(t)=

5 A,t’, such

one of the coefficients A,= A_(l b 2) is irrationAl.‘Then the function of the function e 2iw(r) (i is the integral part of r), is function of the 8 set g&V= lIei<, (1 -PI), where llel 1 [3,6]. If the function e2in+‘(r)is regularized^by a convolution with a real function K(t) EL,, which gives u,(t)= fi e 2im(al)*K (t), the autocorrelation function of u,(t) becomes g,(B)= cug,(dl)*(K(B)*K(B)), with Jla g,(B)= K(O)+K(O) 13961.

RANDOM FUNCTIONS WITH GIVEN TIME COR~LATION

143

Consequently, g(8) being an experimental autocorrelation function, it is sufficient to determine a function K(6) with K(@)*K(S)=g,(f?), to obtain a velocity fluctuation u,(t) having an autocorrelation function g,(e) as close as required to g,(e). For that, it is sufficient to put K= 9??‘-‘)((tg)‘/2), where 9 is the Fourier transform. u,(t) must be a real function. Therefore take u,(t)= fi ReeZi”+‘(%*K(t) (Re signifies real part of), where the factor V?! comes from the fact that taking only the real part divides the correlation by 2. For simulating the maxima functions I,(.$), start [I] with a sequence of all different, where 6a is as small as wanted. values ff,+, E 1)L - sot ,ly + 2 2 [ I Then the space-time correlation function of the fiction

k,-0

k,-0

is

If a and (Z,,WJ are taken great enough, and for chosen sequences (a&u@, g(rc, IC”;0) tends, at an approximation as good as required, to

and it is sufficient to take the convolution of this with (K(e))‘2 to obtain a space-time longitudinal correlation field as close as required to g,,(&,B).

STARTING POINT OF THIS PAPER

It is interesting to calculate, beyond the correlation functions, the asymptotic distribution functions [7] of the theoretic turbulent speeds, in order to compare them to the experimental values of those distribution functions. That difficult problem has not been tackled directly. Previously it was randomized or pseudorandomized. A pseudorandom function is a nonrandom function. It is easier to replace it by a pseudo-“random function,” having a similar correlation function.

JACQUES MAURIN

144 PLAN OF THIS PAPER

What has been written previously leads to the progress report below: Define the notions of pseudoprobability, equidistribution, serial and time correlation functions. Recall the definition of Weyl’s polynomials and the statistical properties of their values. Recall the way in which Weyl’s polynomials generate pseudorandom functions, which are nonrandom functions. Pseudorandomize a sequence of Weyl’s polynomials. Adapt the use of a randomized or pseudorandomized sequence to the generation or simulation of random functions having a previously given space-time correlation function. Calculate the stochastic distribution function of the random functions thus generated, which is a nonrandom function. Estimate the asymptotic distribution function of the trials of those random functions. Bases of This Paper PSEUDOPROBABILITY

Let X, be a sequence of independent r.v. (random variables) having the same distribution function, or, equivalently, of independent trials X,, of a r.v. X. It is known [8] that the distribution function of the results xx of those trials, defined by F,(x)=&

2

x,
1,

XEZ,

tends to the distribution function F(x) of the r.v.; i.e., 1

E

2A+ 1

c

X*
lDzFfm=F,(x) --B

z F(x)DG’Pr(X
where L means a stochastic limit, defined by f’“, Pr(]F,(x) - F(x)] > S) = 0 -f for any 6 > 0. Let, then, xx be a sequence of numbers having arbitrary origin. If, as A+cr, 1 x 1 tends for any x, except perhaps on a set of F*(x) = 2R+1 XA
RANDOM FUN~IONS

WITH GIVEN TIME CORRELATION

145

measure zero, to a limit F(x), that limit will be defined as the asymptotic distribution function of the sequence (xJ. 1 of xx less than x is obtained by adding, x Let x’ > x. The number X&
Because distribution distribution defined by

2 1. Xx<% -n
the function F(x) so defined has the properties of a stochastic function, therefore any property related to r.v. connected with functions only, transfers to the p.r.v. ~seudorandom variables) the sequences (xx).

Pseudo stochastic independence

...,N_I~h,N~h), nEZ Let (x~)=(-NXA,-N+IXA,...,nX~, vectors having an asymptotic distribution function

be a sequence of


fi

F,(x,),

which means that the limit of the ratio of the number of

n--N
mxA

JACQUES

146

MAURIN

EQUIDISTRIBUTION

A sequence equidistributed

of numbers x,,x2, . . . ,x,, . . . ,xA, X E N*, x ER, is said to be on [0, l] if for any interval [a,b] c[O, 11: l=b-a.

That definition is generalized to a sequence of vectors x,,x2,. . .,x,, . . . ,x,,; x, E Rk. That sequence is said to be equidistributed on the unit cube C, = [0, Ilk if for any interval P=[ P=[a,,b,],...,[a. ,f 6.1 1 ,-*-9 [ak,bk]cCk],

lim _L A-Km h

)= XxEP

l=

fi (bi-a,). i-1

AChCA

Then the scalar sequence x1,x2,. . . , xA, . . . , x,, is said to be k-equidistributed on [0, l] if the vectorial sequence xi, x2, . . . , x,, . . . , xA where xA = x~+~__,), is equidistributed on C,. (x*,xh+I,...,xh+i,..., In the same way any vectorial sequence x1,x2,. . .,x, ,..., xA, where x,= vector, is said to be k-equidistributed on C, (1XA,ZXX, * * * > I xh) is a r-dimensional if the vectorial sequence &,G, . . . ,&, . . . ,(,, where

is equidistributed on C,. It is possible to prove the propositions

below. 5,9,10.

In order that the sequence (xx) be equidistributed and sufficient

that for

any function

h(x)

on [0, I], it is necessary

Riemann-integrable

over

[0, 11,

f’rn= i $ h(xh)= /‘h(x)dx. + In orAr ‘that the s&en, (xJ be equidistributed on [0, 11, it is necessary and A

sufficient that for any nonzero integer I, lim 1 2 eZinlxA = 0. A+a Ax-, Let xA = A_& where + denotes the fractional part of a. In order that this sequence be equidistributed on [0, 11,it is necessary and sufficient that A be irrational.

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

147

Those Propositions are generalizable to vectorial sequences. In order that the k-dimensional sequence (xA) be equidistributed on C, = [0, Ilk, it is necessary and sufficient that for any function h(x) Riemann-integrable over ck,

x’ERk.

In order that the sequence (xh) be equidistributed on Ck, it is necessary and sufficient that for any vector I =(I ,, . . . , lk) with integer components not all zero, jmm+ +

$

e2inPxk=o_

Let the’ constant vector A = (A,, . . . , A& and let x,, = Ax 7 (A@, A&...,A&, , . . , A& In order that the vectorial sequence (xx) be equidntributed on C,, it is necessary and sufficient that for any I E Zk - {O} : A *I4 Z. What is said above is easily extended to a sequence (x~), XE] -A, + A] c Z, A-+oc. It suffices to consider a sequence (h’), x’ E N*, in one-to-one correspondence with the sequence (A), XEZ, that bijection being, for example, defined by X=21X/+ lx<*,

A=(_ l)$j

and to apply the previous results to the sequence A’, replacing -!A $,

by

J-g 2h+1 h--A’ The theorems giving equality between the limit when A+cc and the corresponding integral may be stochastically interpreted as ergodic theorems. Indeed a T.V.~ui~st~buted on [0, lJk has the probability density f (x) = lxECk; and the previous theorems may be written

’

12

_!i&(x+ 2h+1 _*

sss tk) ...

h(x’)f (x’)dx’,

which is the definition of ergodism. Consequently, it is possible below to replace all arithmetic averages, functions of sequences transformed from equidist~buted sequences, by the corresponding pseudostochastic moments.

148

JACQUES MAURIN

Independence of the components of an equidistributed vector

Taking a,=0 gives F(h)=

fi b,, where b=(b, ,..., bi,..., b,J. i-l

The marginal pseudoprobability that ixx < bi is defined to be the limit of the ratio of the number of ixA< bi to the total number of ixx whatever the values of the other components be; when it is only required that they be included in [O,l], then 4-1 ifj#i, and Z$(b,)=b,

Ii l=b, j-1

j#i

It is then possible to identify each bi with &(b,), which gives

f’(b) =

Ii4 (41,

i-1

the equation which is the definition of the pseudostochastic independence of the components Cxh) of the vector xx. Conversion into another given distribution

Given F(Y) it is required that the a.d.f. (asymptotic distribution function) of the sequence (Yx) be F(Y). For that it is sufficient, knowing an equidistributed sequence (xx), to take y, = F(-‘)(x~). Indeed, then x, = F(yh), so that Pr(y,
< F(Y))= F(Y)

which follows from the definition of the equidistribution, in which the ratio of is (b-a): it is sufficient to take a=O,b= F(y).

xx~[a,b]

SERIAL AND TIME CORRELATIONS

The serial correlation function of the sequence of numbers (x,), if it exists, is defined for 6 integral by g(B)=

iim _-L-T A+* 2AR+l _nx,‘xk+i*

RANDOM

FUNCTIONS

WITH

GIVEN

149

TIME CORRELATION

Then let x(t) be a function from R to R, the square of which will be supposed Lebesgue-summable over any finite interval. Its time correlation function, if it exists, is defined for 8 ER by

g(B)=

lim L/‘x(,).x(,+~)&. T-+co~T

_T

It has been shown that” g(O) is even. It has been verified [6] that if x(t) is a step function given by x(r)= x;, and if the serial correlation function g(8) of the sequence (xx) exists, the time correlation function of x(t) is, for 8 > 0, given by

take T= A integral.

Indeed

+A

g(8)=

lim ‘1 A-03 211 _A

Then I_:=

x(t)*x(t+e)dt=

where??$=i+e=i+dif

z

I”’

lili

&s,ld!

t
jrnm $$ +

z

*i’ l--h if ,r>l-0,

h-l

g(8)=

gives

x;*xg&

whence

A-I

x;.x;+i+

z him

-!?x 2A ;__Axi’xi+i+l,

I- -A

As (2A+ 1)/2A tends to 1, the first limit is equal to g(d). As for the second, it is zero since g(B) exists. Consequently, if a sequence (xx) satisfies g(B) = 0 for any nonzero integer d, with g(O)= 1, then for 0 > 0, g(8)= le<,*(l - 0). The sequence (xh) has a square summable over any finite interval simultaneously with the step function x(t). Consequently g(0) exists with g(8) and is

JACQUES MAURIN

150

Fig. 1. even, if it may be written

This is a function whose graph is an isosceles triangle with base [ - 1, + 11and height 1, symmetrical about 8=0 (Fig. 1). Conversion into another given correlation function

Start with a step function x(t)=x; satisfying gX(6)=lle,<,(1-181). It is required to determine a real function y(t) the correlation function of which is g,(e), previously given. That problem is approximately resolved [3,4] in the case of a correlation function having a spectral density 4(v) > 0,$(v) EL,, i.e., is integrable, verifying

where g,(0) is real and even, and consequently so is q(v). Then y(t) is pseudorandom; that is to say, it has a correlation function g,(e) tending to 0 when k-+00. To pass from g,(e)= llri
lim r e-2in’fK(t)di, where “2 means that it is a r--PmJ -r

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

151

quadratic mean limit, defined by T__I_+Il~(p)-~_~-l’~r~(~)~~12~~=o. lim K(t) is real, except perhaps on a set of mezsure zero. If K(~)EL,,

the real function y,(t) = a’12.x(cur)* K(t) = (~~1~ +m~ (as) / K(t - s)ds has a correlation function ,g,(e) satisfying :I% .gY(e)=~Y~Q. If K(r)BL,, ,&r)=a1/2x(ar)*&-(r), where KT(r)=l,,,
A Weyl’s polynomial [5] is a polynomial cp(Q with real coefficients, and at least one coefficient other than ~(0) irrational. It has been shown [5] that when AEN, the sequence of numbers &A) is equidistributed modulo 1. In other words their fractional parts xx = cp(X)are equidistributed on [0,11. But those sequences are not k-equidistributed for any k. It has been shown [12] that in order that the sequence (xx) be k-equidistributed, it is necessary and sufficient that k < p, the degree of the variational coefficient of the polynomial. That is one of the reasons which led to the pseudo-randomization mentioned in the Introduction. However, it is shown that for any 8 integral, the 2-dimensional sequence (xx) =(x,,x,+;) is equidistributed on [0, 112. Correlation function of the sequence &A).

It is g(4)=

lim - l zXAxh+& h+m 211+1 __h __ Since the vector x=(x~,x,,+~) is equidistributed on [0, 112, the ergodic theorem mentioned above may be applied. Setting h(x’) = xixi+i consequently gives, if 1 < 6: g(B )=

JJ1”x~x;+sdx~dx;+B.= a, o,o

and setting h(x’) = xi2

152

JACQUES

MAURIN

w~)=l~u~<1u-I~o is required as the correlation function of the function x(t)=x;, it is necessary to transform x, into the form 1,=2fl (xx-t), equidistributed on [- 67, + V/7]. Then indeed, setting 9R . = it gives

%x,

=

lim --!A-Pm 2A+1

$ xA = /‘x;dx;=f, _A 0

and

=12(9RX~‘XX+&-#lLX~+9Rx~+~)+~)=o and Xh- i))‘) = 12(Xx,2 - ?-xx, + a> = 1.

g(O)=S((2fi(

Simulation of Random Functions With a Given Time Correlation Function [13] It has been shown above how to construct a function y,(t), the convolution of a step function, the correlation function .g,(0) of which tends when (~-xc to a given correlation function g(0) having a spectral density. Generally, it is easier to calculate the distribution function of the result y,(t) by pseudo-randomizing it. For that purpose, the single polynomial q(h) is replaced by a sequence of polynomials q,,(X) pseudo-randomly independent Modulo 1. The irrational coefficient other than q,,(O) of any one of them is denoted by A,,. PSEUDORANDOM POLYNOMIALS

INDEPENDENCE

OF A SEQUENCE OF WEYL’S

PROPOSITION 1. Let 7) be a transcendental number. Take the A, in the form a,@‘.-+ b,,, where a, #O and b,, are integral, and the exponents p, are nonzero integers all different. Then the fractional parts ,,xh = q,,(A) X E N >> of the Weyf’s polynomials are pseudostochastically independent. It has been seen that it is equivalent to say that for any N, the vector on the hypercube xX=(-N& - N+iXA,***,&,*.*,N-1 xh, Nxx) is equidistributed C ZN+l’ For that, it has been also shown that it is necessary and sufficient that )mm

-+

+

$ h-l

e2inh=

0

for any IEZZN+‘-

(0).

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

153

This is the case if y, = I-x,, is a Weyl’s polynomial, because then, e2er being periodic with a period equal to 2a,

and consequently yA is equidistributed on [0, I]. Now, in I-x,,,=

2 I;cp,@), the irrational coefficients of a power of X n--N equal to or greater than 1 may not be zero. Indeed, suppose that Q of them, numbered n,, n,, . . . ,nQ, refer to the same power. In I-x,, if for brevity a$ is denoted aq, the coefficient of that power is written

5 (a,$‘q+ b,). Now q,

being transcendental, is not a root of any algebraic’equation with integral coefficients. Then the coefficient of the power concerned cannot be zero. Hence, being transcendental, it is irrational, and so I-x, remains a Weyl’s polynomial, which leads to its equidistribution modulo 1 on C2N+,, whatever be N. A way of getting p,, all different is to take p, = I,,,*n + 1,&n - 1). It is also possible to replace the a, and b, by rational numbers. Indeed, it suffices to multiply them by their lowest common denominator to find again that ?I is not a root of any algebraic equation with rational coefficients. Thus the size of each “active” coefficient a,~“+ b,, can be altered so that the vafues v,,(A) are immediately dispersed along the entire interval [O,I] but not more, in order to avoid useless jumps in the calculation. If, for instance, n is taken equal to e = 2,718.. . ,Napier number, if the relevant degree for cp,(A)is equal top = 2, and if pn is taken equal to 2, it will be 1 I sufficient that u,,e*2*is near to 1, then a,, is near to 4.2,7182.. . = 29,556, * * ’ and that b,, is zero. a,, may be taken equal to I/30. SERIAL CORRELATION

FUNCTION OF THE SEQUENCE

It has been seen that the independent p.r.v. nxh=@ may be treated as a true independent T.v., meaning that it is possible to treat their asymptotic distribution function as a true distribution function. Consequently, everything is said below is equally applicable to a sequence of r.v. or of p.r.v. The r.v. or p.r.v. the X-rank trial result of which is xA is designated by X, with, for a p.r.v., because its ergodism, %X= EX. Let then (X,) be a sequence of independent real T.v., having any distribution functions. Suppose that the means EX, = p,,“, EX,‘= A,“, EXi= ~lq,~exist.

154

JACQUES MAURIN

$

SetyNW= &

the sum of r-v. Then gigge r.v. y(k)= jim, y,(k).

XCX~+~,which is in general a T.v., because it is correlation function of the sequence (X,) is the

-+

PROPOSITION 2

p+ exists, and if

2 tl==-N

then the correlation function of the sequence of independent r.t). (X,,) satisfies y(O) z MO (stochastic limit). if for k#O,

Mk = iirn* &

+

exists and if

lim --!_2N+

N-ax

i n--N

Sl,#l’lll,fl+k

1

for % = t h,2,n- ~~,~)(~*,~+* - &,+A 4 = &(~Lz,+c - liL+k) and % = &J, then the correlation function of the sequence (X,) satisfies &+k(P2,ny(k) L Mk.

In order to show this, start from the statement below of the weak law of large numbers:

LEMMA 2.1 Let (Y,),,,

be a sequence of r.o. satis~~ng the condit~~ below.

Their means El’,, = h exist, with lim N+m

1

m-i-1

2 n__N

p”=M*

Their variances u,’ = u2( Y,) = E ( Y,, - p,J2 and covariances r,,, = E ( Y, - p,J ( Y, - &, n J” I, exist, with

N

Then 2, = &

x

n--N

Y, satis~es lim 2, g M.

N-+x?

It is necessary to show that for any 6 > 0, JirnWPr(lZ, - N 1;b 6)=0. For +

this, the Chebyshev inequality is used.

155

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

2

EZN=&

5

EY, -A

n--N 2,

-

ES!&

A. Then Y;= Y,-cl, $

centered T.v. satisfying Z&= &

are

and Zh=

n--N

YL,with

n--N

a*(Zk)=

EZ;=

E

(2N-t 1>2

n--N

which gives

02(Zi)=

I

s

EYL2+

5

EY;Y;

n,l= -N

The Chebyshev inequality applied to Zh is written, if p = S/Za(Zh):

0 Q Pr(lZh/ 2 pu(Zh >) < --+,

i.e., 0 Q Pr( AZ;/ > 4) <

4a2(z;) 62

.

Putting A‘= ZN - EZ, and A’ = EZ, - M, the inclusion below between domains of the plane (A’, A”)

gives

But under

the conditions

iim EZ, = ?im,&

N-8.%

3

Consequently i< ,““, PriiiN +

of the statement bk = M,

then

MI > S) ;CC; 0.

f” +

Iim 4-

N-3W

82

=0,

and

Pr@?Z&.- Ml > I, = 0.

156

JACQUES

MAURIN

Proof of Proposition 2.

Case k = 0. Then

y(O)= jirnW &

5 X,“. Set Y,,= X,‘. Under -t nm-iV conditions of the first part of the Proposition, The Y, are independent, then the r,,,=O. The EY,, = EX:=

pzvn exist, with trn-

5

&

-f

The

a:( Y,,) = E(Xj

Consequently

Mm

= p+

- &

exist,

with

Lemma 2.1 applies and gives

Then y(k)=

&q

lim -

jirnW & *

5

n--N

-_

an equation

=

n--N

- ~_l~,~)~ = EX,4 - ptn

y(O)=

Case k#O.

~2,n

the

resolvable

_$

X,X,+,.

Form

n--N

(xl - P*,n)(&+k- !%n+k)

kn+kXn+

2~+1

$

XnXn+k,

in h

i n__N

,

hn!hn+k

9

which tends to y(k)

if that

II--N

limit exists, which is the case and gives the value of y(k)- if all other terms have limits. Limit of the left-hand side. Put XA= Xn-p,,n, and Yn=XiX,‘+k. The Xi

being

independent

with

the

X,,,

EY, = EXiEXL+,

= 0,

whence

The Y; = Y,,- EY, = Y, are pairwise orthogonal, i.e., r,,, = 0. Indeed, EY,’ Y,’ = EY,, Y,= EXiXL+kX;Xi+k, I# n. If 12 n + k, each of the r.v. on the right-hand side of the equation above is independent of the other; then EY,’ Yi= EX,‘EX,‘+,EX,‘EX,‘+, 30.

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

157

If 1= n z!zk, for instance, I= n + k, EY,' Y;= EX,'EX;2EXi+k, whence, the three r.v. on the right-hand side being independent with EX;= EXi= 0,EY,' Y/

= 0. In any case, it gives E YJY/= E Y,, Y,= 0,i.e., r,, = 0. Finally, the X,l being independent,

whence, by expanding,

In short, under the second condition of the second part of the proposition, The Y,l are or~ogonal, whence in the Lemma r,,=O. The EY, = 0 exist, whence The

lim - ’

N-ma

2NM

EY,=O.

5 *_._N

u,“(Y,) exist, with

Consequently, the

lim Z,= _lilim&

2

N-GO

Y,

n-‘-N

(&- P*,n)(Xm+k - El,,n+k)

Limit

of

&

$

$m.& -*

ii

2

EY,=O.

n--N

on the right-hand side. Set Y,,= pi,nX,,+k.

h,n-%+k

n--N ‘hen

EYn

i-%,n+k)*?

=

h,nEXn+k

when=,

by

=

and

h,nh,n+k~

expandhit,

a*(

Y,)=

~2(Yn)=C:,nu2(Xn+k)=~~,nE(X,+kt&t

jli,n+k

-

d,,+k).

In short, under the third condition of the second part of the pro~sition, The Y, are independent with the X,, whence rnr= 0. N

The EY,, = &np,,n+k

Mist,

with z”, +

z_N’ii

x n---N

EY,,= Mk.

JACQUES MAURIN

158

2 u,Z(Y,) =o. The c&Y,) exist, with lim 1 N-tcQ2N+ 1 i n--N Consequently the lemma applies to the second term on the right-hand side, and gives

1

N

x F,,~+~X,, on the ~~t*hand side. It suffices in the Limit Of 2N+ 1 n__N reasoning above to commute n into n-t-k, whence for that term, under the fourth condition of the second part of the proposition, the same stochastic limit Mk. I N 2 ~,,~‘~~,~+k.It iS by hypothesis &. Limit Of 2N+l n__-lii Returning to the initial equation, it becomes, in fact, because of the previous results, y(k)=

1 lim $ XnXn+k N-Pm 27?+ 1 n___N

’

O~~k+~k-~k=~k.

It is easy to give examples satisfying the conditions of the Proposition. It suffices, for instance, to consider a sequence of normal centered r-v., with ELrn=u,‘= 2cos22mx, q irrational. Then psn never again takes the same value, so that the’ distributions of the X, are all different. The sequence nn being equidist~buted modulo 1, because of its ergodicity, gives

with Mk =O, whence y(O) g 1 and y(k) 2 0 when k#O. Making time correlation trials with zero pseudoprobability

If X-O, the sequence of the polynomials q=(A) reduces to that of their constant terms tp,{O), which may be arbitrarily chosen. It is, for instance,

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

159

possible to obtain a time correlation function constant and at the same time to have the values p,,(O) discretely equidistributed [ 141. In order to avoid that, it is preferable to take h E N* = N - (0) or A= Z* = z- (0). TIME CORRELATION

FUNCTION OF THE CORRESPONDING

STEP FUNCTION

It has been shown that the time correlation function on 8 ER of a nonrandom step function x(t) = x; is given by

g(O=dJ w-B)+d+

1%

for8 >O

If g(6) = 1 when 8= 0 and 0 when 4 > 0, taking into account the symmetry, it reduces to g(~)=l~e~~,(l-l~l)‘ Here the step function is random, i.e., X(t)= X;; but the same rule applies to each of its trials, which, if MO-- 1 and Mk =0, gives by identifying k to 8, with 8 2 0,

whence by y(d) i li_e and by symmetry, Y(e)

TIME CORRELATION

zl,e,
FUNCTION OF THE CONVOLUTION

The randomization above is transmitted to the convolution. Since the Bass method applies to each trial of the random convolution, that method is stochastically applicable to it. More precisely, it is shown below that the convolution is a weighted sum of T.v., and Proposition 2 is by hypothesis applicable to the sequence of them. Consequen~y, to obtain a r.f. (random function) Y(t) the time correlation g,(B) of which is given with a spectral density q(v): Make H(v)=$‘/‘(~)e @(*)E Ls whatever cp(u) odd may be. Determine Kit, such that H(p)=%_(K(t)) in L,. If K(r) EL,, the random correlation function ,yy(@) of Y,(t) = cri/2X(~t)*K(t) satisfies J$im.,y,(6) 1 gy(@).

160

JACQUES MAURIN

If K(t)@&,

the random

correlation

function

.,=yr(8)

of Y&t)=

CX’/~X(~?)*K,(~), where K&f)= l,,,
The r.f. Y,,=(f) so obtained are real, as are the r.v. X;. They do not tend to any limit when (Y+OO.

Distribution Function of the Convolution of a Step Random Function [15]

In order to compare with experiment, it is important to know the d.f. (distribution function) of the function _Ya(t) or Y,,,(r), which, putting-t= at, may be written Y, k =X(i)ta’/‘K( ( 1 Replacing

Ya( i)

or Yo,=(k)

i)

or Y.,,( i)=X(i)*~l/~K,(

by Y(t), a1i2K( k)

i).

or a1j2KT( i)

by

K(f) E I,,, and omitting the tilde, the problem restricts to calculating the d.f. of

the r.f., Y(t)=X(t)*K(t).

What is written below is abbreviated by putting i= 7. PROPOSITION 3 Let X, be independent r.v. havingp.d. (probability densities) f*(x), and the set of absolute moments E IX,.,1bounded as n ten& to infiniv; and let the convoluting function K(t) be summable: Then Y(t) has a d.f. almost evevwhere differentiable Q (y, t) given, at each of its continuity points, by

Q(y,t)=~~~e,(y,t),e,
f-’ e,W~Ody’>MO=

In+” K(t-s)cis

v+N

x

n

f(xJ.dx,_N...dx7_,.dx,+1...dx7+N.

n-7-N ?I#7

qN(y,t) is thep.d. of Y,(t)=

x

k,,(t)*X,.

n-7-N

Indeed, Y(t) = / ‘“X(t).K(t-s)cis= --oo

‘f k,,(t).X,= n---o0

jirn- Y&t). +

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

Let y be the result of a trial of YN(t). Then y =

x

161

&(2)*X,.

n-7-N

In the space (x,_~, . . . ,x7_ ,, x,, x,,,, . . . , x,,~), go from the basis (X+_N,~.~,XI_I,~~,X~+I,...~X,+N)to the basis (x,-N,...,X~-I,Y,X,+,,...,X~+~). 7+N

The elementary probability

fn(x,)d$_N ’ - - dx,_ ,~Ix;dx,+~ - . * dx,+# is

n n-7-N

expressed here by multiplying it by the modulus of the Jacobian of the inverse transformation, which gives

74-N x

f,(x,)dx,_N,..dx,_t.dy.dx,+l.*.dx,,N.

II:

“XGN r+N y=

x

k,,(t).x,

gives x7=-&

y7

n-7-N

I

T5N k,(t)*x, n-r-N ll#T

. i

Indexing the Jacobian’s rows by i and its columns by j where i,j= r Iv ,..., 7)..., r + N, and writing S,, the Rronecker symbol, the Jacobian may be written

Consequently the elementary probability becomes

T+N l-T

whence, integrating on dx,_N,. . .,dr,_ ,,dxT+,,. . .,d~2+~, one obtains the expressions of the p.d. qN(y,t) and of the d.f. Q,,(y,t) of Y,(i) in the proposition.

162

JACQUES MAURIN

According to the theorem of Levy [16], it then suffices to show that the sequence of the characteristic functions &(~,t), the Fourier transforms of the q&,t), tends to a limit K(v,t) continuous at v =O; then its inverse Fourier transform Q(u,f) is the d.f. of Y(t). Omitting t in the functions and z?zcc in +m, it * gives s -03

TfN

x

u

f,(x,)dx,_N.‘.dx,_,.dx,+,...dx,+N.

n-7-N ?I#7

Returning to the expression x, = i 7 [.y-- 8:$x$

then

shows that the integrand is summable in y; that integration carried out the integrand is clearly summable in the x,,n=T-N ,..., r- I,T+ I,..., T+N, for s

f,(xJdx,

= 1. Finally, the functions f, are positive. Consequently, by the 74-N

Fubini-Tonelli theorem [ 171,the integrations commute, and in y =

I:

k,,x,

n-7-N

changing the variable y+x,

gives

I+N

where the “a are the Fourier transforms of thef,. The convergence of K,(v) is consequently that of the infinite product in the

163

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

last equation. It is known [18] that in order for the product to converge absolutely, i.e., in order that its value be independent of the ordering of the r+N

factors, it is necessary and sufficient that this also happen for

z:

n-7-N %-

-9 -1. If i is differentiable, by ?&(O)= fs,(x,)&,,

I4

= 1 the mean value theorem

gives G”(k&= 1+ k,&;(Q), iE[O,v]. Now, in order that f, be differentiable, it suffices [t9] that f,(x) admit a first-order moment EX,,, which is implied by the existence of the E]X,]; since, furthermore, the set of the E[X,,] is bounded by a limit ?& the derivatives are bounded. Indeed, 9; (v) = 2in

J

e2iv”z xf,(x)dx gives I%*(P)]< 25~ ]x]f,(x)dx= f

27rEIX,J <2&h. Then tu,l=IG~-ll=Ik,vr~(k,i))~f~~~lvllk,f;

then the convergence of

T-t-N

KN

is equivalent to that of

2

IQ. But

n==v--N

because K(t) is summable. The ]k,J being positive, it implies [ZO]the conver7i.N gence of 2 ]&I, whence that of &,. n-7-N

It remains only to show the continuity of the limit K(v) at Y==O. First, each u,,= Gn(knv)- 1 is continuous at v =0, for on the one hand s(O)==0 because j%(v)] < 2d&/k,~~ vI, an d on the other hand Iu,(v) - u,,(O)]= Iti,( Q 2nG~]k,,]]v] tends to Oyith v. Now the infinite product

n

Gj,(k.v ) converges uniformly on v E [ - I,

+ 11,Indeed, it has been seen thl;$conv:rges,

interval, lu,(v)l<2~&F~lk,l= Cl,, with

and, on the?tJher hand, on that

2 u,,==277‘?; 2 lknl where the n=-m ?I----cI)

+g ]un(v)\ converges ?I=--oo independently of v, i.e., uniformly, and then so does the product &(v). From the continuity of each u,,(v) at v = 0 and the uniform convergence of r+N T+N f&(v)= n G_(k,v)= fl (1 +u,,(v)) on [- 1, + l] 30 follows [21] the bounding sum on the right-hand side converges. Then

n-c--N

~-T-N

continuity of the infinite product J&(Y) at v=O.

JACQUES MAURIN

164

Then the theorem of Levy [ 161implies Proposition 3. COROLLARY 3.1 Let there be a point where Q,‘(y, t) exists. Then, if the sequence f,(x,,) is such that qN(y, t) is continuous if N > NO,and if, on a segment

[yo,yI,qdy’, 0 conuergesuniform&to dy, 0, then QJv, 4 = dy, 0. Indeed, first, omitting t in the variables, if q&y) is continuous, qN(y) also is for any N > NO. Indeed, qN,+,(y) may be written

T-EN

x

fl

f(x,Jdx,-N...dx_,.dx,+,...dX,+N.

n-7-N ?l#T

Moreover, the reasoning used in the proof of Proposition 3 shows that

r+N,,

x

n

f,(xn)dx,-N,...dx,-,.dx,+,...dx,+~,

is the p.d. of pNO=

x n-T-(N,+l)

X,,, which gives

RANDOM FUN~ION~

”

i.e., setting &._(N~+lf=k+-(N~+DXI-(N*+l) 1 4&7(Y)= k&Jo+i) If =

165

WITH GIVEN TIME CORRELATION

,

a~WLcNo+r,=f

(zr-(No+I~)~No(Y-&-(No+I))%4vo+1)

’ 1) (hNo)(Y)' k-_(No+

fis summable, and by hypothesis qNocontinuous. It is known [22] that their convolution ri2v,is continuous. since q@,+i(Y)= ~~+~o+,(~,+%+l)~~o(Y- ~~+~o+~~~~+~o+~)dx,+~o+~~ the same reasoning shows that qNo+, is continuous, By induction it follows that qN is continuous for any N > Ne. Consequently, by a well-known theorem, q(y), the uniform limit of continuous functions, is continuous. Finally, Jirnw s ‘qN(y’)&‘= lYq(y’)&.

Indeed, q&y’), being continuous

for N 2 Ne octhezgment [Ya,Y],‘&,by a well-known theorem, bounded there, i.e., IqR(y’)l s; Q,. As it is possible to write lq(y’) - q&y’1 < l in the form of MY’) - e < q(Y’) < qN(Y’)+ e, it becomes QN- c < q(Y’) < qN+ e or MY’)] Q Qiv+(:=Qm, and q(y’) is also bounded on [y,y]. It then follows, from the Lebesgue theorem 1131,that lim J ‘qN(y’)&‘= J’q(y’)&‘. Consequently it N-xo Yo Yo is possible to write

Finally, Q (ye) being a constant and q(y’) being summable and continuous upon [ye,y], it follows, from a well-known theorem, that Q(y)= q(y). COROLLARY 3.2. Oppose that the hypotheses of Proposition 3 hold and that the set of the second-order moments p2,n= EX: of the independent r.v. X,, is bounded, Then, for the convolution Y(t) to be a normal r.f., it is necessary and sufficient that each X,, be normal. T-+N

Indeed, then the variance of Y,,,(t) is Si =

x

k,f(t)u,2 where u,”= ~a,~-

n-7-N

is the variance of X,,. Under the conditions of Proposition 3, F,,~ is bounded with E]X& and under those of Corollary 3.2, pz.n is bounded. It

d,,

JACQUES

166

MAURIN

7-l-N

follows

that

u,’ is bounded

by

a&, whence

Si < u&

x

k:(t).

But

n==~-N K(t

it gives Sl~(r-s)lds(lK(t-u)ldU<

summable,

ik

00

.2

c(1

n+l’lX(t-s)p 1”

1

The two terms on the left-hand which gives

F

k;(t)<

tl=-co

side each being positive, they must be finite,

2 (X”+“lK(t-S)la)*
Consequently,

being

m

n

I

?I---00

K(t)

co, i.e., consecutively

~“+*‘lK(t-s)jml;“‘L’~K(t-S)ldu<

n&n=-ma

+m

f

k:(t) G

whence

- s)dr

the variance

/N

of Y,(t)

does not tend to infinity;

then that

2 1

of Y(t) is finite. This then proves [24] Corollary

ERGODICITY

3.2.

OF THE CONVOLUTION

OF THE STEP RANDOM

FUNCTION

[25]

Experiment gives, not the d.f. of a r.f. of the time t, but the a.d.f. (asymptotic distribution functions) of some trials of it. Consequently it is important to know how go back from the a.d.f. to the d.f., which we call s.d.f. (stochastic distribution functions) in order to distinguish them from the a.d.f. The r.f. Y(t) being constructed as previously by a convolution between a summable function K(t) and a step random function X(t)= X7 obtained from

RANDOM FUNCTIONS WITI-I GIVEN TIME CORRELATION a sequence of independent r.v. X,,, define the s.d.f. of X(t), designated by F(n,t)= the s.d.f. of Y(t), designated by Q(v,t); the a.d.f. of X(t)

and

Y(t):

167

F,.(x);

E(x)=

lim -!=1 T_.,m2T I _= ytr)(v +dt, f?(v)=

lim 1 * 1ylrl_,vdz, where on the right-hand side there are Riemann r-+m 27- f _T stochastic integrals [26]; under the existence conditions which will be used below, fl and Q are r.f.; the m.d.f. (mean stochastic distribution functions) of X(f) and Y(t): P(x) = Jirnm& I_’ F(x,t)dt,

G(Y)= $mm & I_’ TP(y,t)dr. -+

If they exist,

they are ~onrando~ functions. The present paper concerns the transition from E and 0 to F and @, and conversely. PROPOSITION 4.

If F(x) exists, g(x) 1 F(x).

g means that when

1

Tl x(0-D dt tends stochastically to F(x) T+oo ’ 2T _= I

[271. To prove the proposition, it suffices to apply, for x determined, the weak law of large numbers to the r.v. V,,= l,fl..x, independent with the X,. They have: a stochastic mean EF”=O*Pr(X, > x)+ 1 *Pr(X,, < x)= F,(x). a variance E(~~-E~JZ=E2k’##(l-F,(x))C(l-Ei/,)2F,(x)=F,(~)(lSuppose that F(x) = Jim, +T $_’ F,(x)dr exists. Then, if T is restricted to + T its integral values N= f, the partial sequence N-1

1 J 2N

N F ++

(x)dt=&

N-t

c ln+‘F,(x)dt= n=-N a

2N _2N+l has the same limit. FN(X) 2N+l tending to 1 and 2N+1 2N

2N+l 1

&

r:

F,(x)-&(x) ( n__N 5

< &-+-i

F,(x)

n--N

1

to 0 consequently gives

168

JACQUES MAURIN

Moreover, F,(x)( 1 - F,(X)) being bounded by Q, 5 E(Vo-EVJJ2
whence E(V,-

EiQ2


v2;+

1 =O.

Consequently the sequence (V,,) verifies the conditions of the weak law of large numbers, which is written here

Since V, = lXmCn= may be written

n+l fn

N-l

I,,,

dt, the left-hand side of the expression above

n+l

I,(,),,&

It is easy to verify, as above, that the last expression may be written dt= P(X), which proves Proposition 4. PROPOSITION 5. Under the conditions of Proposition 3, if the can~iuting function K(t) has u gouged support, and if e(y) exists, then 0 (y) % Q(y). The proof of Proposition 5 starts from the two lemmas below.

LEMMA 5.1. Let V(t) be a r.f. satisfying

s T

V(t)dt g

-I:

lim ‘f’EV(t).dt. T-GO~T

_T

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

169

The proof below is due to Bass. Start from the equation

’ )

= E

(V(t)-

EV(r))dqT

(V(f)-

EV(t’))dt’

*

-T

Tbe right hand side may be consecutively written 1 -j-j-“’ 4T2

1 4TZ

E((V(t)-EV(t))(V(t’)-

EV(f))dtdr’)

-T,-T

(E(v(t)V(t’)))dtdr’-

1

4T2

It is easy to verify that if lim a(T) exists, that the limit of the expression?za> be written as

The limit of the right-hand side of the initial equation being zero according to the statement of Lemma 5.1, the value of the left-hand side is also zero, which gives 2

V(t)d#-

&JT

EV(t)dt -T

=o, )

an expression which may be written

quadratic mean limit proves the lemma.

where

&e

m.q.

=

implies the stochastic limit L , which

170

JACQUES MAURIN

LEMMA5.2. Let Y(t) be a r.f. Besides its a.d.f. Q(y) and its m.d.f. e(y) define

its

m.b.d.f.

(mean

bisector

distribution

function)

g(y)

=

lim _.I_“’ S(y,y; t,t’)dtdt’ where S(y,y’; t,t’) is the j.d.f. (joint T-too 4T2 J-J -T,-T distributionfunction) of the pair of r.u. Y(t), Y (t’). Suppose that e(y) and s(y) exist, and that s(y)=

g2(y).

Then &(y) e e(y).

In order to show that, apply Lemma 5.1 to the r.f. Y(y,t)= and t given, it is a Bernoulli T.v., satisfying

1r(rj(v. For y

EV(y,t)=O~Pr(Y(t)>y)+l~Pr(Y(t)
For y, t and t’ given, the correlated Bernoulli r-v. V(y,t), Y(y,t’) are not zero and equal 1 only if both Y(t) < y and Y (t’)
1

= lim

2T

T+m

so that Lemma 5.1 can be applied, giving

s T

V(y,t)dt

-T

Proof of Proposition 5.

such that this support =

n+lI

I ln

K(t-s)dr

f

JimW&J’EV(y,t)dt=e(y). -T

The support of K(t) being bounded, there exists A is included in [-(ljI), +(#- l)]. Then k,,(t)

is zero as soon as It-n-l,r-n]z[-(IS-l),+(H-l)],

which occurs if either t-n<-fi+l or n>t+&-1, or t-n-l>fi-1 or n < t - 15. The first condition is implied by n > 7 + I?, the second by n < r - I?. Consequently k,(t)=0 if n4[7-#,r+H]. Then let t < r’ be two instants such that 7’ - r > 2151,which happens if t’ - t > 2# + 1. Then the largest value of n such that k,(t) #O is n = T + #‘; and the smallest value of n such that k,(t’)#O is n’ = 7’ - & > 7 + fi = n, so that none of the independent r.v. X, being a component of Y(t) is also a component of Y(r’), and conversely. Consequently Y(r) and Y(t’) are independent T.v. as soon as t’- t > 2fi + 1. And likewise for V( y, t) = 1r(tj
RANDOM

FUNCTIONS

WITH

GIVEN

TIME CORRELATION

171

tion 5; and that if E ( V(y, t) V(y, t’)) = S (y,y; t, t’) exists, so does s(y) according to the statement of Proposition 5. As soon as (t’- tl>2&+ 1, V(y,t) and V(y,t’) being independent, S(y,y;t,t’)= Q(y,t)*Q(y,t’).Besides $J/

S(y,y;

t,t’)dtdt’

Ir'-tl<21S+l

1 +4TZ Since S(y,y; 2T(4N+

t, t’) < 1, the first integral

2); then its quotient

Since S(y,y; t,t’)= sion on the right-hand 1 2~

S(y,y;t,t’)dtdt’.

by --&

on the right-hand side if less than 4fi+2 which tends to 0. is less than 2T,

Q(y,t)Q(y,t’) when It’- tJ>2H+ side may be written

1 r r _ TQWWt.ZT I _TQWP'= /

1, the second expres-

1

,~,_l,<~n;+,Q(y,').Q(y,f')d'd' 4T2 J/f

where the last expression, the quotient of a constant divided by 4T2+ca, tends to 0. Because the limit of the first for T+ca is e2(y), and the limit of the left-hand side is g(y), therefore s(y)= e’(y), so that Lemma 5.2 applies and gives Q(y) L Q(y). PROPOSITION 6. Under the conditions of Proposition 3, the r.f. Y(t) has a j.d._f.

given by

+Y(y,y'; t,t’)=

(7'-1+2N-I)

... T’+

k4t’)y

- k,(t)y’+

N

IX

x,,n(t9t’)xn

n-7-N Rf7,7'

II

T’+N - k(f)Y

+ k(t)y’+

x

Xz,(fT4X”

n=7-N fl27,+ r'+N

x

fl

f,(x.)dx,_N...dx,_,.dx,+,...dx,_,.dx,,+,...dx,,+N.

n-7-N n27.7'

k,(t)=0f0rnE[7+N+1,7’+N],k,(~‘)=0f0rnE[7-~,7’-N-11.

1

172

JACQUES MAURIN

s(y,y’;f,f’) is the j.d.p. ofthepuir YN(?), Y,(f). has been‘designated by t’, i.e., t’- t > 0.

Thegreater ofthe two rimes t,i’

In order to prove this, the proof of Proposition 3 is extended to two dimensions [28]. Below, t and t’ are omitted, and k,, is written for k,(t), ki for k,,(f), and Y, for Y&t), Y; for YN(t’). First it is established that the independent T.V. X,, appearing both in T’+N riiv 2 k&x,, satisfy nE[r’-iV,7+N]. YN(t)m x k,X, and Y,,,(t)= n-r’-N

n-7-N

To obtain

go from the basis (x,_~, . . . ,x,_,, x,, x,+g, . . . , +.._I, to the basis (X,_Nt‘,‘tX,_1,y~,x,+,,...,x,,+l,.*.,y’,x,,,,

a,,

x,~,x,~+~,...,x,,+,)

. . ..x.+.&

with 7,-l-N

k;.y- k,,y’f

2

x,,(t,t’)x,

n-7-N I##T,7’

=-

r'+N

1

xT’ x+.+(f,f‘) where, in x,,(t,t’)=k,k~,-k~k,,, nE[7-N,7’-N-l].

1

-k:v+k,y’+

r,

i

k,=O for ~E[T+N+~,T+N]

and k;l=O for

++N

Then the elementary probability

n[

fn(x,)dx,_N * - * dx,.+N is expressed

n-7-N

by multiplying it by the modulus of the Jacobian J of the inverse transformation, It is sufficient to integrate that probability on x,._~, . . . , AZ,_,, in order to obtain the j.d.p. of the pair Y,, Y,& X r+l,‘.‘,X,‘-I,X7’+I,...rX,‘+N Setting ij=r-N ,..., 7,..., T’,..., T’-t-N gives

8,

J=------- 1 X,44 0

( 1- aiT)( 1- hd 1 +si,((l-6,)(1-S,,)x,,(t,t’)+6~k:,-S~k,,) +~i~~((1-5’,)(1-8~)x,,(t’,t)--~k;+6fik,) ij

where %n (f, t’) = -x,+(6 9,

and x,&‘, j) = - x~,,(t, t’) are satisfied. Since

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

173

x,+( t, f’) = k,k; - k: k,, expansion gives

Mul~plication by /JI and inte~ation over the x,, n = r- N,.,.,T- l,~+ 1,...,7’- l,r‘+ 1, . . . , T’ + I?, results in the expression of s, given in the statement of Proposition 6. From then on its suffices to show that the sequence of corresponding characteristic functions, Fourier transforms of the s~(v(Y,_Y’), i.e., &(B, v’)= I$

ezin(~+“‘y’)sN(y,y’)dydy’, tends to a limit K(~,Y’) continuous at v,v’=O.

Replacing x, and x,, inf, and&, by their values as functions of y and y’, it is possible to write

X

(T’--T+zN-

JJ

1)

I f,(y,y’;x,_N

,...,

tX;,-,,X,,+I,...,X,,+N)

X*-~,X++1,...

I’+N

x

n

f,(x,)dx~_N’**dx,_,.dx,+,...dx,_l’dx,’+,*..dx~,+N.

R=T-N tS#l,l’

r’+N

In the integrand J,f+ inversely y,y’

n f,, regarded as a function of y,y’, reexpress n-7-N ?Z#7,1’

as functions

Irl~f.(%,)f,(x*)dx~~~,, and x,, x,,. The double

of x*,x,.

Then

JJ

f,(.Y,_v’)f,~(.YJ’)& 4Y’=

where I= ~((~‘~)) is constant with respect to y,y’ 7) 7’ integral on the right-hand

side being written

Sf,(x,)dx,Sf,,(x~,)dx~~= 1, th en the one on the left hand side exists, and then the integrand is summable in y,y’. Once that integration has been carried out, the integrand is clearly summable in the x,,n=r-N ,..., r-1,7+1 ,..., 7’-l,~‘+l,..., T’+N. Consequently, as the functions are positive, the Fubini-Tonelli theorem [17] implies that the

JACQUES

174

7’ + N

r+N

integrations

can be permuted.

Finally

y=

MAURIN

k,,x, and y’=

x

x

k;x,,

n-7,-N

n-7-N

give I= k,k:,- k;k,,= l/J. Then replacing y and y’ by their values above gives

KN(v,v’)=JJ

I Ii I

r+N

exp 2in v k,x, + kTex,.+ I

t

+

(T’--1+2N-

X

I)

c

r’+ N

x n-7’-

k;x,,

N tl#T.T1

I

N

T’+

J

I

PZ#T,T’

v’ k;x, + k;.x,. +

...

k,,x,,

2 ?l=7-N

n

f,

(xn)dx~-N

n=~-N

n=r-N

J

?l#7,7’

n

KN(v,v’)=

g,,(k,,v+

k;v’)

where the ‘3,, are the Fourier transforms of thef,. The convergence of KN(v,v’) is consequently that of the infinite product in the last equation. It is known [18] that in order that the product converge r’+ N absolutely, it is necessary and sufficient that this also happen for z I%L n-7-N

un=rn-l. As before, the E IX,1 existing theorem gives

and being bounded

T,,(knv+k;v’)=l+(k,,v+kiv’)%;(k,,F+kp’) elinvxxf,(x)dx, whence I gives I~~I=Ir~-lI=I(k~v+k,:v’)G~(k,I+k,:I’)l

with $i (v) = 2h

by 9’;, the mean value

where (~,~)~[O,vl~[O,v’l I’%;]< 2n

/

Ixlf,(x)dx

< 2n’3,,

~2~~~(lvlIk~l+Iv’Ilk~l).

which

175

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION It has been previously shown that

converge,

which implies the convergence of .~~]u,,],

hence that of nI[_%,,(k,.+

&Y’), hence that of K,( v, v’). It remains only to show the continuity of the limit K(v,v’) at v=O, ~‘-0. Now each u,(v,~‘)=T~(k,~+k~v’)1 is continuous at (v,v’)=O: Indeed, first ives l%(V~V’)l G27%AIwll+ IV’IIKI)g u,(O,O)= 0, whence on the other hand Iu,,(v,v')~(O,O)~=]u,,(v,v')~<2n~~~~v~~~~~+~v'~~~~~) tends to 0 with (v,v’). Moreover, the infinite product

a

Gn(k,v + kiv’) converges uniformly on

(v,v’) E[ - 1, + 112.Indeed, it has &&nmseen that it converges, and+zoreover, on

that

domain,

/u,,(v, v’)] < 2n %k(Ik,J+ [kJ)= u,

with

x u, = #Z---o0

2nQ m( Z PA+ #---co ir %I)< 00because of the integrability of the conn--C0 vopiting function #(t),+%hich is the definition of the uniform convergence of l-f (1+ u&,v’))= ‘-%rn

II

?$(K,v+ Kn’v’).

th”+gontinui;; of=each a,,(~, v’) at (v,v’)=O and the uniform conver-

gence of

E Yfi on [ - 1, + I]‘EO there results [21] the continuity of the n---a3 infinite product at (v,v’)=O. It follows from the theorem of Levy [28], generalized to two dimensions, that the &(v,v’) converge to K(v,v’), and thus the S,(y,y’; t,t’) converge to S(y,y’; t,t’) at any point of continuity of that last j.d.f., which proves Proposition 6. PROPOWTION 7. Swpose that the hypotheses of Proposition 3 are satisfied. For any N, take for Q’,(y,t) the value (&(y-0,t)+QN(y+0,t))/2, andfor S,ly,y’;t,t’) the ualues (S,(y-O,y’-O;t,t’)+S,(y+O,y’-O;t,t’)+S,(y0,~’ + 0; t, t’) + S,( y + 0,~’ + 0; t, t’))/4. Suppose that S,( y, t) converges uniform& in t on ItI> (Y with a < CO,and that S,(y,y’; t, t’) converges uniformb in (t,t’) on It’-tl >p with p (Y,S 1y,y’, t, t’) tends to the product Q (y, t).Q (y’, t’). In other words, the r.u. Y(t) and Y(t’) are mutually indepe~ent in limit. The principle of the proof below is to take It’ - I]> 2(N + l), so that YH(t) and Y,(t’) remain constantly independent. First, Q&y,t) and S,(y,y’; r,t’), being monotone for any N, have only discontinuities of the first type [29], so that, according to their definitions in Proposition 7, Propositions 3 and 6 apply for any y,y’, i.e., QN(y,t) tends for any y to Q(y,t), and S,(y,y’;t,t’) tends for any pair (y,y’) to S(y,y’;t,t’). Put A&(Y,~)= CMY,~)-- Q and AS,(u,r’;t,s’)=S,ty,y’,r,t’)S(Y,Y’; f,N).

176

JACQUES MAURIN

The convergence being uniform in t, S’with 1t I> cy, 1t’l < a, 1t’ - tl> j3, under these conditions: There exists N(e,y) such that lAQ,(y,t)l< e for any N > N(r,y); conversely, given N, IAQ,(y,t)< c(N,y), a positive decreasing function of N tending to 0 as N-+eo. There exists N(Z,y’) such that [AQ,(g’,t’)l N&,y’); conversely, given N, IAQ,(y’, r’)j < e(N,y’). There exists N(G;y,y’) such that ]AS,(y,y’; t, t’)f N (is,y,y’); conversely, given N, IAsS,ly,y’; t, f’)l p, and on the domain ltf > (Y,lr’l > o[, take It’--tl>Z(N+l), whence 7’--7>ZN+l. Then ~+NGT’-N-l, and consequently none of the basic r.v. X,, is both in YN(t) and in YN(r’). Thus YN(t) and YN(f’) are independent, which is written S,(y,y’; t, t’) = Qnr(y, t) * Qnr(y’,t’). Putting J (N,y,y) = Sup(e(N,y),c(N,y’)), a positive decreasing function of N tending to 0 as N+co, it follows from the definitions of the AQ, that Q(Y, 0 - S < Qhr(y>0 < Q(Y, 0 + S; Q(Y’, W < Ql;(v’, 0 < Q(Y’, t’) + S; whence, by multiplying the first double inequality by the second,

whence, setting + = 21+ I2 and since 0 sr;Q s; 1,

#/IN;y,y’) is a positive decreasing function of N tending to 0 as N-W. Consequently, under the conditions of Proposition 7,

i.e., IS(y,y’;t,t’)-

Q(YJ)-Qtr’,t>l

~E(N;Y,Y’)+O+ICI(N;Y,Y’)=E(N;Y,Y’).

&N;y,y’) is a positive decreasing function of N tending to 0 as N-+co. This proves Proposition 7.

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

177

PROPOSITION 8. With the hypotheses of Proposition 7, if Q(y) and s(y) exist, then Q(y) s Q(y). Indeed Lemma 5.2 applies. To show this, put

whence S(y,y;t,t’)dtdt’

-2T

1

1 r r Q(Y,f’)dl’ I _T I _TQ(y,Gdr*D

The right-hand side tends to s(y)- @(y). If the left tends to 0, the condition of the Lemma 5.2 is fulfilled. Let N be chosen as in the proof of Proposition 7. Divide the plane of integration (t,t’) into a domain 33 where both Iti and It’/ < LYand It’-- r/ < 2(N

Fig. 2.

178

JACQUES MAURIN

+ 1) with 2(iV+ 1) > /3, and its complement e: In 9,]AS] G 1 gives < --&(~~L.~T+ZLY.~T+~(N+ In c?‘,/ASI Q &WV)

2(a+N+ T

1)2T)=

1)

+ 0. t-m

gives

as small as required however large T. Consequently, the ant-hand side is as small as required for N sufficiently large, so that at the Emit S(y) = c2(y) and Lemma 5.2 applies, which proves Proposition 8.

Abbreviated Remits Remember that to obtain, from a T.v. X equidistributed on [O,I], a r.v. Y having a given d.f., it suffices to take Y= F(-‘j(X). A Weyl’s polynomial v,,(A) is one such that at least one of its coefficients A, other than q,,(O) is irrational. Then the sequence rp,(X) the fractional part of r++,(A), is equidistributed on [0, l] when AEZ. Let (q,,(A)), n ~2, be a sequence of Weyl’s polynomials. The A, are given by A,, = u,qfi -t b,, where q is transcendental, the a,, bn are rational, and the p,, are nonzero integers that are all different. Then the .x,=w are p.r.v. pseudostochastically independent. The results below apply both to r.v. and to p.r.v. Let (X,) be a sequence of independent T.v., such that the moments EX,,= N

~~nJ=tf=

IJGW~‘=

~4.n

exist. If ?I&= lim --!._ N-KZJz?+l

2 pz,n exists with n__-#

n_ _* ~2,~- F:,~) =O, then the serial correlation function of the sequence

(X,)

satisfies y(0) g Me, stochastic

for 8 =( fi2.n -

,$,,A

-

P:“+& P2,g+k then

Y@)

= Mk.

-

!a

%

if, for k #O,

1 lim 2N+ 1

I N I?.&= lim x ~i,~*p~,~+~exists with N-Bm2N-6 1 n_ _-N ~bd~2.k

limit;

N-wm =

P~,n~CIZ,n+k

-

,Et:“+k),

@*

=

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

179

It is easy, by a linear transformation of the X,, to obtain sequences verifying M,= 1 and &=O. It has been shown that the time correlation function of a step r.f. X(f) = X; satisfies, for B > 0, y(B)= y(@)(l- e)+ y(6+ l)$ & is the integral part of 6. If y(0) L I and ~(4) 2 0 for 8 >0, then y(e) 2 lyel
,,ry,(@) g g,(e).

Start from a sequzz: of independent r.v. (X,,) having p.d. and the set of the E/X,1 bounded. For simplicity, put i= or, Y,(L/o) or Y&i/a) = Y(~,~1~2~(~/~) or a 1/2KT(f/a) = K(fl E L,, and suppress the tilde. Then Y(t)=X(f)+K(t) has a d,f. Q(y,t), and the pair Y(t),Y(t’) has a j.d.f. s (u,.Y’;$7t’). Let F(x,r) be the d.f. of the r.f. X(t)=X; where the (X,) satisfy the above hypotheses. Define the a.d.f, of X(t) and Y(t) by P(x) = lim Jr 1 dt and Q(y) = lim 1 IT lu(,)(v dr, and their m.d.f. by r_.,* 2T s _ r X(r)
and Q(Y)= Jirnm& J_r rQ (.~,t)& then -+ If F(xT exists, >;X) g F(x); If K(t) has bounded support, and if Q(y) exists, Q(y) L Q(y);

F(x)=

F(x,t)dt

the d.f. of Y,v(t), and S,(y,y’; t,?‘) the j.d.f. of ( YN(t), YN(f’)). Then, if &(y,t) converges ~iformly in t for ItI > a! with (Y< 00, if &&,y’;t,t’) converges uniformly in (t, t’} for ir’ - rl ;b /3 with B < ty, and if Q(y) and ~(y,y;~,t~)d~d~’ exist, then Q(y) I e(y),

The property o(y) 2 Q(y) is an ergodic property. The last result above extends that property to the cases where the support of the convoluting function K(t) is not bounded. The above shows how, by means of stochastic or pseudostochastic trials, a r.f. having stochastically a given time correlation function, and satisfying the ergodic property mentioned above; can be obtained. EXAMPLE. It is convenient to verify with an example that it is possible to obtain a r-f. Y(t) having the properties specified above.

180

JACQUES MAURIN

The example used here consists in taking the independent r.v. X,, normal such that EX,, = 0 and EX,f = 1, and the convoluting function K(t) given by K(t)=.&‘l. It is known that then Y(t) is also normal [27] if the variance 2; of Y,(t) tends to a limit X2. To simulate the r.v. X,, above, it suffices to take ,,x,= F(-*)(tpn(A)), cp, (A) pseudostochastically independent Weyl’s polynomials, with x -+dr. F(x)=& e I Verify that Th: d.f. &(y,t) of Y,(t) converges uniformly in t in some domain Iti > a, and that the j.d.f. S,(y,y’;t,t’) of the pair (j+,(t), Y,,(f’)) converges u~formly in (r,r’) in some domain (r’- r) > f3. Then b(y) s g(y). First show that IjimmZ, = I: exists. It is easily seen that 22, = + i+N

EY,&t)=

x

k:(t) where k,(t)=(e-

A n-t-N

l)e-lrl.en, which gives

and 53e-2!+e-zM

p=

. )+(2-(e-!+e-“-!)))*

with a separation

tending to 0 as N+co. That separation may be made small enough so that AZ,=ZZ, may be approximated by

where e(e-2~+e-2(1-!9 e+l u(t)=

2

~(e-2!+e-20-‘)

((

_ ))+(2-(e-r+e-(1-!))2)“2)

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

181

is periodic in t with a period equal to 1, and continuous, and hence has a maximum uM; AX,(t) < aMe-2N tends to 0 when N+oc. Moreover Q(y,t)=

$J”‘(‘)cz -Y’%,

-z2/2dz gives by differentiating under the with a maximum

!.$Z =-.Nmay ’ za I d= IM

be taken great enough so that ]AQhl]= lQN- Q I may be approximated /AQJ#

g I

AEN< s.

by

In order to have \AQN/< c, consequently it

I

@hi XV2S.E’ QN(y, t) uniformly converges, not only in t, but also in y. verify that there exists p < 00 such that for It’ - II> /3, the j.d.f. t, t’) of the pair (Y,,,(t), Y,(t’)) uniformly converges in (t, t’) to S (y, Here, the pair is still normal for any N [30], and

suffices to take N > N(r)= 4 Log Then Next S,(y,y’; y’; t, t’).

p=dO=

q

Y(t)Ytt’)) x2’

, the correlation coefficient.

It is known that as p+l, which is the case if t’+t whence Z’-+X, the probability concentrates itself on the bisector of the first quadrant of the plane (y,y’) (Fig. 3) [31]. Then it is easily verified that

limp= 6-d

1, with

limS(y,y';t,f)=: P-d

~~i1”‘i”f’y’y”e-~2~~d,

vz

-*

and that the derivative in p of S (y,y’; t, t’) increases indefinitely as p tends to 1. Therefore it is convenient to assign to the separation 1t’ - t I a lower bound /3 < 00 in accordance with the statement of Proposition 7. Differentiating under the integral gives easily

whence

lA~s,i#l~

I

ABN <

enough so that ]AE,SN] < 43,

and it is possible to take great ]Ax,SN]< c/3.

182

JACQUES MAURIN

Fig. 3.

Also, writing Z=y/Z,

Z’=y’/X’

gives

l~l~~((p+~EIZz~I+~(EZi+EZ”))
with

p
The right-hand side being a continuous function of p in p E [0, 1- a] for any a > 0, it is bounded there by S, = $,(a). a may be taken so close to 1 that S,(a) is as close as required to the value of the riot-hand side given by p=O, i.e., E\ZZ’I. It remains to calculate the variation Alp, =pN -p between the correlation coefficients of (Y,, Y,&) and (Y, Y’). It is easily verified that :+N

p;v=Z,Z~p,=EY,Y~=

x

k,(t)~k,(t’).

II=?-”

The following equations may also be verified: For i= 2, p+(2-(e-!+e-(‘-!f ))(2-(e-t’+ e-(-?)) +e-1 --(e-(!+!3+e-((

1- !)+o-f’)))(

1_

e-2N);

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION

183

For t^< ?-- 1, (e- l)e-I(?pb=(e-

+e-o-Q

I)e-(2-i)

i- l)e!-t’

2-(e-f+e-

(‘-!‘))+e-!‘(2-(e-!+e(i-t)))

( -(!+!‘)+e-((l-~)+(l-!‘)))(l_~-2(~-~~-~)))~

+A@

i whence for N--+oo, (e- l)e-‘(r?-{p,_.(e_l)e-(i_i)

l)e!-t’

+e-(1-!)(2-(e-!‘+e-(‘-!.))+e-L.(2-(e-!+e-(1-!)))

.

ft+t’)+e-((l-!)+(‘-~))))

+A(,-

Because here k,,(t) > 0,O < pEy< p' gives Apk = p’ - pb, i.e., as soon as ? - i> 1, A& b; p’. In the equation giving p’, the first term of the bracket is bounded by (e- l)(?- ;). It is verified that the second is bounded by 4(1 -e-‘/2}, and the third by s, e-‘/3+

s)e-(i-i)

Consequently, for any N, A&<(e-l)((e-I)(?--$+4(1tends to 0 as ? - i increases indefinitely. - t) tends to

0 as i’ - i increases indefinitely.

pas-.&givesdp=!?.-xx

p’ iii2 P’dZ’; 2x2’2 I

22r:

whence, AZ, and Xx, being non-negative, a bound of the appro~mation

of

APN,

Ap;
Consequently

which tends to 0 as It’- tl tends to infinity. Then there exists a real number j3 such that if It’- tf > /3,1A&J < 613. Then if N > N(r), as soon as It’--tl > p,

which satisfies the conditions of Proposition 8. There is still uniform convergence in I and in y.

184

JACQUES MAURIN

REFERENCES 1. J. Bass and J. Maurin, Correlations spatiotemporelles dans un fluide turbulent. C.R Acad Sci., Serie A, 266, 376 (1968). 2. J. Mauri& Simulation pseudo-aleatoire d’une turbulence derriere grille. Note Technique O.N.E.R.A, No. 20/2669A, 92320 Chatillon 1968, France. 3. J. Bass, Fonctions stationnaires. Fonctions de correlation. Application a la representation spatio-temporelle de la turbulence. Ann. lust. Henri Poincare, Section B, V, NO. 2, 135 (1968). See also, Stationary functions and their application to turbulence. J. Math. Anal. App. 47, No. 2 and 3,354,458 (1974). 4. J. Maurin, Si~uiation deterministe du ksard, Masson, Paris., 1975, pp. 102-104, 115-120. 5. H, Weyl, Ueber die Glei~hver~~ung von Zahlen Module Ems. M&z. Amden, 77, 313 (1916), 6. J. Bass, Les fonctions pseudo-al&atoires. Mem Sci. Math. CLIII, (1962). 7. Cf. A. Tortrat, Repartition asymptotique des fonctions presque-periodiques de Besicovitch. Proceedings of the Prague Conference on information Theory and Random Functions, Academy of Science, Prague, 1964. 8. Cf. H. Cramer, Mathematical Methods of Stutistics. Princeton Univ. Press, Princeton, 1961. 9. J. W. S. Cassels, An Introduction to Diophantine Approximation. Cambridge Univ. Press, Cambridge, 1957. 10. J. Bass, Suites uniformement denses, moyennes trigonometriques, fonctions pseudo-ale atoires. Bull. Sot. Math. France, 87, 1 (1955). 11. J. Kampe De Feriet, Sur f’analyse spectrale dune fonction stationnaire en moyenne, CoIloque Intentions de M&mtque, Poitiers, 1951, pp. 317-335. 12. J. N. Franklin, Dete~~tic sim~ation of random processes. Mutk Cotnp., 17,28 (1963). 13. J, Maurin, Simulation de fonctions akatoires a fonction de correlation temporelle don&e. C. R Acad. Sci., S&e A, 278,629 (1974). 14. J. Maurin, Suites deterministes i repartitions et correlations anormales. C. R. Acad Sci., Serie A, 279, 625 (1974). 15. J. Mauriu, Densite de repartition dune fonction aleatoire ou pseudo-“fonction aleatoire” en escalier. C. R. Acud Sci., Serie A, 270, 1108 (1970). 16. P. Levy, Calcul des Probabilites, Gauthier-Villars, Paris, 1926, p. 125. 17. Cf. J. Bass, Math.&uatiques III, Masson, Paris, 1971, p. 136. 18. Cf. G. Valiron, Thhorie des Fonctions, Masson, Paris, 1955, p. 53. 19. Cf. J. Arsac, Transformation de Fourier et theorie des distributions, Dunod, Paris, 1961, p. 29. 20. Cf. L. Schwartz, Methodes ~t~~ti~es pour les sciences physiques, Hermann, Paris, 1961, p. 14. 21. Cf. H. Delange, Gours &Art&e, p. 36-37, Fast. 2, Orsay, 1967. 22. Cf. L. Schwartz, Ref. 20, p. 125. 23. Cf. F. Riesz and B. Sz. Nagy, &ons d’AnaIysefonctionnelie, Gauthier-Villars, Paris, 1961, pp. 36-38. 24. Cf. H. Cramer, Random variables and probability distributions, Cambridge Univ. press, Cambridge, 1970, p. 60. 25. J. Maurin, Propriete ergodique d’une classe de fonctions aliatoires. C. R. Acad. Sci., Serie A, 280,673 (1975). 26. Cf. A. Blanc-Lapierre and R. Fortet, Theorie &s fonctions aleatoires, Masson, Paris, 1963, pp. 89-90.

RANDOM FUNCTIONS WITH GIVEN TIME CORRELATION 27. 28. 29. 30. 31.

Cf. A. Blanchpierre and R. Fortet, Ref. 26, pp. 62-63. H. Cramer, Ref. 24, p. 105. Cf. J. Bass, Marhemuriques, Vol. I, p. 123, Masson, Paris, 1961. H. Cramer, Ref. 24, pp. 111-l 14. Cf. J. Bass, Probabifites, Masson, Paris, 1962, pp. 84-85.

Received June, 1976

185