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Stationarity of independent sequences
STATISTICS& PROBABILITY LETTERS Statistics & Probability Letters 23 (1995) 9-11
ELSEVIER
Stationarity of independent sequences B. Lacaze Laboratoire d'Electronique, 2 rue Camichel-31071, Toulouse Cedex, France
Received December 1993; revised February 1994
Abstract
Let A(t) = ~n~z A,#n(t) be a real random function. We propose a necessary and sufficient condition for A(t) to be stationary. The real r.v.'s A. are of zero mean, of unit variance and independent, and the/~n are continuous real functions such that/~n(n) = 1 and #.(k) = 0, k # n.
1. Introduction Let us recall that the A = {A(t), t e ~ } random function is stationary (in the strict sense) when, for any n, the {A(t~ + u), A(t2 + u) . . . . . A(t. + u)} n-dimensional law is independent of u. In the theorem in Section 2, we study the stationarity of the random function A defined by
A(t) = ~ An/~n(t), t~R, n~Z
where (CO An, n 6 Z , are (mutually) independent real variables, with the same probability law such that E(An) = O, E(A 2) = 1. (C2) #.(t) are real functions, continuous on R and obey the conditions:
2. Theorem 2.1. Let (P1) (P2) that
#n(t) = # o ( t - n) V t e R , n e Z , 3 s(og) taking the values 0 and 1 and such
1 ~+~ #o(t) = ~ J-oo ei°ts(co)d°9' s(~o + 2kn) = 1 V~o~R, keZ
/G(n) = 1,
(P3)
#n(k) = O, k e Z , k # n, /~2(t) < oo,
This last condition is necessary and sufficient (Renyi, 1966, p. 392; Lukacs, 1966, p. 83) for the A(t) quadratic mean and almost everywhere existence.
teR.
(1)
n~Z
0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 4 ) 0 0 0 8 8 - P
Ao is normal.
A necessary and sufficient condition for A(t) = Y.,~zAn#n(t) to be defined as a stationary
10
B. Lacaze / Statistics & Probability Letters 23 (1995) ~ 1 1
random function is that the PI, P2, P3 conditions are satisfied.
On the other hand, according to (C2):
,f+;i_
#o(n) = ~
2.2.
The P1 condition is necessary. It can actually be written, following (1):
_ e
dto,
(7)
n~_Z.
The unicity of bounded measure (Rudin, 1960, p. 17) then implies:
transforms
doto(to) = ~ dS(to + 2nrt) = dto, E [ A ( t ) A ( t + z)] = Y' #.(t)#.(t + z),
(8)
n~Z
(2)
n~Z
given that the A, are zero-mean unit variance and independent (Cl). In order for A to be stationary, it is necessary for (2) to be independent of t. In particular, for t integer and t = 0, according to (C2): E [ A ( t ) A ( t + z)] = #.(n + z) = #o(Z),
(3)
hence (P 1).
which shows that S(to) is absolutely continuous and that its derivative s(to) = dS(to)/dto satisfies: otb(to) = ~ s(to + 2mr) = 1.
(9)
It remains for us to demonstrate that s(to) can only take the values 0 and 1. We first note that, according to (5), (6) and (9), ot~(to) is absolutely continuous for any relR. Then, according to (5), ICdto)l ~< oto(to) = 1, and ei°*ot'r(to) is periodic with regard to ¢o of period 2ft. Finally, by arranging (3) and (5):
2.3.
The P2 condition is necessary. First of all, the #o(Z) continuity (C2) makes it possible to apply the Bochner-Khinchine theorem (Feller, 1955, p. 586; Loeve, 1955, p. 207):
1 [+oo (4)
#o(Z) = ~ j _ ~ e i'°' dS(to),
where S(to) is real, nondecreasing and such that S ( - oo) = 0, S( + oo) = 2ft. Relation (4) can be written in the following way: #o(Z) = ~
_ e i''' d0t,(to),
d0t,(to) = ~ e 2irtk' dS(to + 2kr0,
z~.
(5)
//n('r) = / / o ( ' t " - - n) = ~
l
_
e-i°~nl-ei°~ot',(to)]
dto.
(10)
Relation (10) expresses the fact that #o(Z - n) is the nth Fourier coefficient of e i'°' ot'~(to). To this function, the Parseval equality can be applied (Rudin, 1960, p. 27):
1 [+Z / ~ g ( z - n) = ~ j _ . I¢~(to)l 2 dto.
ill)
n6g
As the A(t) random function is assumed stationary, we have E[A2(t)] = E [ A 2] = 1 VtelR, hence, according to (10) and (11): 1= ~
_
I¢,(to)l 2 dto.
(12)
k~Z
In particular,
f+" e i¢°ndoto(to), ~o(n) = ~1 j_. doto(to) = dot,(to) = ~ dS(to + 2kn), kEZ
neZ.
(6)
We have seen above that Iot',(to)l <~ 1. Then (12) implies that, for any z: ]ot',(to)] = 1 almost everywhere with regard to co (for the Lebesgue measure on ( - n, + ~)). Lastly, for almost every to, ot'~(to)is a characteristic function in the probabilistic sense, according to (5). More precisely, it is the characteristic
B. Lacaze / Statistics & Probability Letters 23 (1995) 9-11
function of a random variable taking the values 2krt, keZ with the probability s(co+ 2kit). The fact that I~'~(~o)1 = 1 almost everywhere, implies that it is degenerate (Lukacs, 1960, p. 19). So, for each co, (5) becomes: 0(r(co) = e2ink('°)*S(co + 2kn(co))
(13)
with
s(co + 2jr0 -- 0,
for j :# k(co),
s(co + 2rtk(co)) = 1. This shows that P2 is necessary. An alternative possibility for the proof can be deduced from (Llyod, 1959).
11
By applying the Parseval identity (Rudin, 1960, p. 27):
~, lto(t
-- n)#o(t
+
z -- n)
n~Z
=
- -
2/t
e
-
itoz -
2ink
(tolt
dco.
_
Thus, E [A (t)A(t + z)] does not depend on t. Then the random function A is stationary in the wide sense. The P3 condition allows us to affirm that A is a Gaussian random function. Indeed, the Levy continuity theorem (Lukasc, 1960, p. 48) implies that for any n, the random vector (A(t), A(t + zl) ..... A(t + z,)) is Gaussian; in this case, stationarity in the strict sense is equivalent to stationarity in the wide sense.
2.4. The P3 condition is necessary. It is immediately deduced from Laha and Lukacs theorem (Laha and Lukacs, 1965; Lukacs, 1968, p. 116). /ao(t) is actually continuous and then takes all [0, 1] interval values (C2). As a result, for a certain to value of t:
A(to) = E a.A., n~Z
where at least two of the a. are not zero and where ~.~z a2 = I. It is enough to ensure that the law common to the A. is the normal law.
2.5. P~, P2 and P3 a r e sufficient. P~ and P2 e n s u r e stationarity of A in the wide sense. In fact, /to(t ) : ~
_
e ion+2ink(t°)t dco,
k(co)eZ.
Now, ~to(t- n) is the Fourier coefficient associated with e i~t+ 2i,tk(~)t.
3. Conclusion
Insofar as E(A.) and Var A. exist, the stationarity (in the strict sense) of the random function defined by A(t) = ~.~z A.#.(t) is a restricting property. At the first order, it imposes a Gaussian character; at the second order and at the upper orders, it confers a particular shape on the interpolation function po(r) = #,(~ + n) which is also the autocorrelation function E [A (t) A (t + z)].
References Feller, W. (1955), An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York). Laha, R.G. and E. Lukacs (1965), On a linear form whose distribution is identical with that of a monomial, Pacific J. Math. 15, 207-214. Llyod, S.P. (1959), A sampling theorem for stationary (wide sense) stochastic processes, Proc. Amer. Math. Soc. 92, 1 12. Loeve, M. (1955), Probability Theory (Wiley, New York). Lukacs, E. (1960), Characteristic Functions (Griffin, London). Lukacs, E. (1968), Stochastic Convergence, Heath Math Mono. Renyi, A. 0966), Calcul des probabilit~s (Dunod, Paris). Rudin, W. (1960), Fourier Analysis on Groups (Wiley, New York).
ELSEVIER
Stationarity of independent sequences B. Lacaze Laboratoire d'Electronique, 2 rue Camichel-31071, Toulouse Cedex, France
Received December 1993; revised February 1994
Abstract
Let A(t) = ~n~z A,#n(t) be a real random function. We propose a necessary and sufficient condition for A(t) to be stationary. The real r.v.'s A. are of zero mean, of unit variance and independent, and the/~n are continuous real functions such that/~n(n) = 1 and #.(k) = 0, k # n.
1. Introduction Let us recall that the A = {A(t), t e ~ } random function is stationary (in the strict sense) when, for any n, the {A(t~ + u), A(t2 + u) . . . . . A(t. + u)} n-dimensional law is independent of u. In the theorem in Section 2, we study the stationarity of the random function A defined by
A(t) = ~ An/~n(t), t~R, n~Z
where (CO An, n 6 Z , are (mutually) independent real variables, with the same probability law such that E(An) = O, E(A 2) = 1. (C2) #.(t) are real functions, continuous on R and obey the conditions:
2. Theorem 2.1. Let (P1) (P2) that
#n(t) = # o ( t - n) V t e R , n e Z , 3 s(og) taking the values 0 and 1 and such
1 ~+~ #o(t) = ~ J-oo ei°ts(co)d°9' s(~o + 2kn) = 1 V~o~R, keZ
/G(n) = 1,
(P3)
#n(k) = O, k e Z , k # n, /~2(t) < oo,
This last condition is necessary and sufficient (Renyi, 1966, p. 392; Lukacs, 1966, p. 83) for the A(t) quadratic mean and almost everywhere existence.
teR.
(1)
n~Z
0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 7 1 5 2 ( 9 4 ) 0 0 0 8 8 - P
Ao is normal.
A necessary and sufficient condition for A(t) = Y.,~zAn#n(t) to be defined as a stationary
10
B. Lacaze / Statistics & Probability Letters 23 (1995) ~ 1 1
random function is that the PI, P2, P3 conditions are satisfied.
On the other hand, according to (C2):
,f+;i_
#o(n) = ~
2.2.
The P1 condition is necessary. It can actually be written, following (1):
_ e
dto,
(7)
n~_Z.
The unicity of bounded measure (Rudin, 1960, p. 17) then implies:
transforms
doto(to) = ~ dS(to + 2nrt) = dto, E [ A ( t ) A ( t + z)] = Y' #.(t)#.(t + z),
(8)
n~Z
(2)
n~Z
given that the A, are zero-mean unit variance and independent (Cl). In order for A to be stationary, it is necessary for (2) to be independent of t. In particular, for t integer and t = 0, according to (C2): E [ A ( t ) A ( t + z)] = #.(n + z) = #o(Z),
(3)
hence (P 1).
which shows that S(to) is absolutely continuous and that its derivative s(to) = dS(to)/dto satisfies: otb(to) = ~ s(to + 2mr) = 1.
(9)
It remains for us to demonstrate that s(to) can only take the values 0 and 1. We first note that, according to (5), (6) and (9), ot~(to) is absolutely continuous for any relR. Then, according to (5), ICdto)l ~< oto(to) = 1, and ei°*ot'r(to) is periodic with regard to ¢o of period 2ft. Finally, by arranging (3) and (5):
2.3.
The P2 condition is necessary. First of all, the #o(Z) continuity (C2) makes it possible to apply the Bochner-Khinchine theorem (Feller, 1955, p. 586; Loeve, 1955, p. 207):
1 [+oo (4)
#o(Z) = ~ j _ ~ e i'°' dS(to),
where S(to) is real, nondecreasing and such that S ( - oo) = 0, S( + oo) = 2ft. Relation (4) can be written in the following way: #o(Z) = ~
_ e i''' d0t,(to),
d0t,(to) = ~ e 2irtk' dS(to + 2kr0,
z~.
(5)
//n('r) = / / o ( ' t " - - n) = ~
l
_
e-i°~nl-ei°~ot',(to)]
dto.
(10)
Relation (10) expresses the fact that #o(Z - n) is the nth Fourier coefficient of e i'°' ot'~(to). To this function, the Parseval equality can be applied (Rudin, 1960, p. 27):
1 [+Z / ~ g ( z - n) = ~ j _ . I¢~(to)l 2 dto.
ill)
n6g
As the A(t) random function is assumed stationary, we have E[A2(t)] = E [ A 2] = 1 VtelR, hence, according to (10) and (11): 1= ~
_
I¢,(to)l 2 dto.
(12)
k~Z
In particular,
f+" e i¢°ndoto(to), ~o(n) = ~1 j_. doto(to) = dot,(to) = ~ dS(to + 2kn), kEZ
neZ.
(6)
We have seen above that Iot',(to)l <~ 1. Then (12) implies that, for any z: ]ot',(to)] = 1 almost everywhere with regard to co (for the Lebesgue measure on ( - n, + ~)). Lastly, for almost every to, ot'~(to)is a characteristic function in the probabilistic sense, according to (5). More precisely, it is the characteristic
B. Lacaze / Statistics & Probability Letters 23 (1995) 9-11
function of a random variable taking the values 2krt, keZ with the probability s(co+ 2kit). The fact that I~'~(~o)1 = 1 almost everywhere, implies that it is degenerate (Lukacs, 1960, p. 19). So, for each co, (5) becomes: 0(r(co) = e2ink('°)*S(co + 2kn(co))
(13)
with
s(co + 2jr0 -- 0,
for j :# k(co),
s(co + 2rtk(co)) = 1. This shows that P2 is necessary. An alternative possibility for the proof can be deduced from (Llyod, 1959).
11
By applying the Parseval identity (Rudin, 1960, p. 27):
~, lto(t
-- n)#o(t
+
z -- n)
n~Z
=
- -
2/t
e
-
itoz -
2ink
(tolt
dco.
_
Thus, E [A (t)A(t + z)] does not depend on t. Then the random function A is stationary in the wide sense. The P3 condition allows us to affirm that A is a Gaussian random function. Indeed, the Levy continuity theorem (Lukasc, 1960, p. 48) implies that for any n, the random vector (A(t), A(t + zl) ..... A(t + z,)) is Gaussian; in this case, stationarity in the strict sense is equivalent to stationarity in the wide sense.
2.4. The P3 condition is necessary. It is immediately deduced from Laha and Lukacs theorem (Laha and Lukacs, 1965; Lukacs, 1968, p. 116). /ao(t) is actually continuous and then takes all [0, 1] interval values (C2). As a result, for a certain to value of t:
A(to) = E a.A., n~Z
where at least two of the a. are not zero and where ~.~z a2 = I. It is enough to ensure that the law common to the A. is the normal law.
2.5. P~, P2 and P3 a r e sufficient. P~ and P2 e n s u r e stationarity of A in the wide sense. In fact, /to(t ) : ~
_
e ion+2ink(t°)t dco,
k(co)eZ.
Now, ~to(t- n) is the Fourier coefficient associated with e i~t+ 2i,tk(~)t.
3. Conclusion
Insofar as E(A.) and Var A. exist, the stationarity (in the strict sense) of the random function defined by A(t) = ~.~z A.#.(t) is a restricting property. At the first order, it imposes a Gaussian character; at the second order and at the upper orders, it confers a particular shape on the interpolation function po(r) = #,(~ + n) which is also the autocorrelation function E [A (t) A (t + z)].
References Feller, W. (1955), An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York). Laha, R.G. and E. Lukacs (1965), On a linear form whose distribution is identical with that of a monomial, Pacific J. Math. 15, 207-214. Llyod, S.P. (1959), A sampling theorem for stationary (wide sense) stochastic processes, Proc. Amer. Math. Soc. 92, 1 12. Loeve, M. (1955), Probability Theory (Wiley, New York). Lukacs, E. (1960), Characteristic Functions (Griffin, London). Lukacs, E. (1968), Stochastic Convergence, Heath Math Mono. Renyi, A. 0966), Calcul des probabilit~s (Dunod, Paris). Rudin, W. (1960), Fourier Analysis on Groups (Wiley, New York).