Extremes of a random number of variables from periodic sequences

ELSEVIER

Journal of Statistical Planning and Inference 45 (1995) 133-141

of istical planning Jst a t O Urnal and inference

Extremes of a random number of variables from periodic sequences Helena Ferreira Department of Mathematics, University of Coimbra (CMUC), Apartado 3008, 3000 Coimbra, Portugal Received 24 December 1992; revised 9 March 1994

Abstract In the class of periodic sequences, satisfying an appropriate long range dependence condition, we investigate the weak convergence of the kth order statistic M~) where 1 ~1 is a sequence of positive integer valued random variables such that N,/n converges in probability to a positive random variable N.

Key words': Periodic sequence; Point process of exceedances; Sample of random size

1. Preliminaries and framework T h r o u g h o u t this p a p e r {X.}.~>I will be a periodic sequence of r a n d o m variables, i.e. there exists an integer T~>I such that for every choice of integers il . . . . . in, (Xi,,..., X~,) and (X~, + r ..... X~, + r) are identically distributed. We shall refer to such a sequence as a T-periodic sequence if T is the smallest integer as above. If T = 1 then {X.} is a stationary sequence. Let {u.}.~>l be a sequence of constants. The long range of dependence condition A(u,) (Hsing et al., 1988) is said to be satisfied by {X.}.~>~ if an,/=sup{ IP(A c~B)--P(A)P(B)I: A~fl](Un), B~flT,+t(u.), k = 1..... n - l }

(1.1)

is such that e.,t,--,0, as n ~ oo, for some s e q u e n c e / . = o ( n ) , fl{(u.) denotes the a-field generated by the events {Xs <~u.}, i <<,s <~j. Taking in (1.1) only the events A = {Xi, <<,u ...... Xip <~u.}, B={Xj,<~u ...... Xjq<~u.}, with 1 ~ ~ l , we obtain the distributional mixing condition D(u.) (Leadbetter, 1974). An account of the theory of extremes for weakly dependent periodic sequences, defined above, can be found in Alpuim (1988). Write M . = m a x l ~ i ~ . X i and S,.[ X,u . ] ( B ) = ~ i =" 1 l{x . . . . }6L(B), for any Borelian B in (0, 1]. m

0378-3758/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 7 5 8 ( 9 4 ) 0 0 0 8 2 - 4

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H. Ferreira/Journal of Statistical Plannin9 and Inference 45 (1995) 133 141

Applying Khintchine's convergence of types theorem, it was proved in Alpuim (1988) that if the maximum M., under suitable normalization, converges weakly as w n ~ ~ (. ~ ) then the limiting distribution is max-stable.

Proposition 1.1 (Alpuim, 1988). Let a . > 0 and b., n>>-1, be constants and H a nondegenerate distribution function such that P(a~ 1( M . - b,) <~x) .~o~ H(x). Suppose that the T-periodic sequence {X.} satisfies the dependence condition D(u,) with u. = u.(x)=a.x + b., x ~ . Then H is an extreme value distribution. If for each ~ > 0 there exists a sequence {u.}.>~l such that n 1-~

P(Xi<~u.) . ~

z

(1.2)

i=1

and P(M.<<,u,)-x~oexp(-Oz), 0~<0~<1, then 0 is called the extremal index of {X,} (Alpuim, 1988). A sequence {u.} satisfying the first condition (1.2) will be denoted by u~. Results in Hsing et al. (1988) concerning the convergence in distribution of S.[X, u.] may be extended in a straight forward manner to T-periodic sequences if one takes r. = [n/k. T] Tinstead of [n/k.] and uses T-periodicity. Hence, the following d necessary and sufficient condition for the convergence in distribution ( ~ ) of S.[X, u,] holds.

Proposition 1.2. Let {X.} be a T-periodic sequence satisfying A(u.), u.=u~.for some z > 0 and {k,} such that k.%,l. ~ o 0 ,

k.l./n .--,~o~ O.

Then the following statements (i) The sequence of point a compound Poisson process Lebesgue measure), 0 < v < ~ , (ii)

P(M.<<,u~.) ~ e -v,

are equivalent: processes {S.[X,u~.]}.>~I converges in distribution to S on (0, 1] with intensity measure vm (m denotes the and with distribution of multiplicities n.

0
(1.3)

and, for each integer s >~1, [n/k.T]r

n.(s)=P

~, i=1

[n/k.T]r

llx,>uz}=s

~ i=1

)

l/x,>u~}>0

~ n(s).

(1.4)

n

This result can also be obtained from general results about convergence in distribution of multidimensional random measures developed in Nandagopalan (1990). We shall denote the compound Poisson process S by S Iv, n]. As in the stationary case, S Iv, n] depends on T only through the intensity v.

H. Ferreira / Journal of Statistical Planning and Inference 45 (1995) 133-141

135

The introduction of local dependence conditions (Ferreira, 1993) for periodic sequences, which generalize condition D'(un) in Leadbetter (1974) and condition D"(un) in Leadbetter and Nandagopalan (1989), has made it possible to obtain (1.3) and (1.4) from certain limiting probabilities, easy to compute.

2. Extremes of a random number of variables

Let {N.}n~I be a sequence of positive integer valued random variables such that

Nn/n converges in probability to a positive random variable N. We shall investigate the limiting distribution of the kth order statistic of X~ ..... XN,, under the same linear normalization as the maximum M.. This problem has been discussed for i.i.d. sequences (see Galambos (1978) and references therein, and Barakat and EI-Shandidy (1990)). The results for i.i.d, sequences will be extended for periodic sequences with extremal index and satisfying a long range dependence condition slightly stronger than A(un). We shall deal with the number of exceedances N. SN,[X, un(X)]((O,b])=~,i=ll~x,>,,(~,))6i((O,b]), where Un(X)=a.x+b., which is N.

related with the above order statistics in an obvious way. Definition 2.1. The sequence {Xn}.>~I satisfies condition d(u.) for a given sequence {Un}.~>X, if

fin,~= sup { IP(A n B)-- P(A)P(B)]: A ~ fir (un) ~ fir (Uk), B ~ fl~ +l(u.),

l <~k <~n-1} is such that fl.,~, .~oo0, for some sequence l.=o(n). The following lemma is a consequence of the condition _A(u.) and Lemma 6.2.1 in Galambos (1978). Lemma 2.1. Suppose that the T-periodic sequence {X.} satisfies _A(u.(x)) and u.(x) = u~{x), z(x) > O. If S. IX, un(x)] n~o Sty) then

P(Sn [X, un(x)] ((0, b]) <<,s, B)n ~oP(S(~)((O, b]) <~s)P(B),

(2.1)

for each event B, 0 < b <~l and non-negative integer s. Proof. By Lemma 6.2.1 in Galambos (1978), if the sequence of variables Sn[X, u.(x)]((0, b]), n ~>l, satisfies the limit relations

lirn P(S. IX, u.(x)] ((0, b]) ~O,

k = 1,2 .....

lira P(Sn [X, u.(x)] ((0, b]) <<,s [Sk [X, Uk(X)] ((0, b]) ~
H. Ferreira/ Journal of Statistical Plannin9 and Inference 45 (1995) 133 141

136

where the convergence is for continuity points of SIX)((0, b]), then (2.1) holds for any event B. For the sake of simplicity in the notation write St.x) for S.[X, u.(x)]. We have

P( S(.X)((O,b ]) <<.s, S~X)((O,b ] ) <<.s) >~P(S~.X)( (O, ~JY])=O, S(.X)((k~j~,b])<~s, S~")((O,b])<~s) =P(S(.X)((~J2, b])<~s, S~X)((O,b])~s)+o(1), since u.(x)= u~(x). For n large and applying A (u.(x)), this last quantity equals

P(S(.~)((~,b])<~s)P(S~X)((O,b])<~s)+o(1) = P (Sp)((0, b]) ~
(2.2)

Using analogous arguments we also have

n(s(.")((O,b])~s, S~X)((O,b])<~s)

= P (S (.x)((0, b] ) ~
[]

The next lemma contains Theorem 2.2.1 and Lemma 6.2.4 given in Galambos (1978) and plays an important role in the proof of the main result in this section.

Let { Y,},~>l be an i.i.d, sequence and u.(x)=a.x +b.. Suppose P (S. [ Y, u. (x) ] ((0, 1] ) = O). ~o H (x), where H is a nondegenerate distribution function. If {N,},~>l is a sequence of positive integer valued random variables such that N./n.~ N in probability, where N is a positive random variable, then Lemma 2.2 (Galambos, 1978).

lim a ~ = A N n ~ ~

aN"

and

lim b N , - b , n ~

~

aN.

Bu,

in probability.

H. Ferreira / Journal of Statistical Planning and Inference 45 (1995) 133-141

137

Furthermore, At and Bt are continuous and monotonic functions of t defined by the relation Ht(x) = H(Atx + Bt), t > O. The idea of the p r o o f of the first part of the next result is close to that of L e m m a 6.2.3 in G a l a m b o s (1978), which states the same result for i.i.d, sequences, since if {X,} has extremal index 0 > 0 then the normalization constants a, and b, are the same as those for the m a x i m u m M , generated by an i.i.d, sequence with c o m m o n distribution F(x) = [It= 1 p1/r (Xi <~x) and P(S [v(x), n] ((0, 1]) ~ s) is a continuous function on x, as occurs for i.i.d, sequences. H o w e v e r the following alternate a p p r o a c h pertaining to point process of exceedances needs some new arguments.

Proposition 2.1. Let {X.} be a T-periodic sequence with external index 0 > 0 and {N.}.~I a sequence of positive integer valued random variables such that N./n ~o~N in probability, where N is positive random variable. If, for each x E ~, the condition _A(u,(x) ) holds and S. [X, u,(x)] . ~ S (x) = S Iv(x), n], H(x) = lim.~ o~P(S.[X, u.(x)]((0, 13) = 0) being a nondegenerate distribution function, then (i) P(SN. [X, UN.(X)] ((0, b]) <~s, B). Z'~ P(S [v(x), n] ((0, b]) <.s) P(B), (ii) lim P(SN.[X,u.(x)]((O,bl)<<.s)=

P(S[zv(x),n]((O,b])<.s)dP(N<.z). --

c~3

Proof. Let t o < - - - < t r "

be real numbers and consider the events Do={N<,to}, Dk = {tk- 1 < N ~ tr.}. Suppose ti's are chosen satisfying P(Do) + P(Dr. + 1) < e with e > 0 arbitrary. (i) F o r large n we obtain

P(SN.[ X, UN.(X)]( (O, b]) <~s, B) N. <<,P(SN.[ X, UN.(X)]( (O,b]) ~S, t o < - - ~ t r . ,

B)+2e

n

= ~ P(SN.[X, UN.(X)]((O,b])<~s, tk-, < N ~ t k , B ) + 2 e k=l

&

n

N.

<~ 2, P(St.,k ~l[X, UN.(X)]((O,b]) <~s, tk-1 < --<<,tk, B)+2E k=l

n

<<-~', P(St.,k_d[X, UN.(X)]((O,b])<~s, tk-1 < N <~tk, B)+3e. k=l

(2.3)

138

H. Ferreira/Journal of Statistical Planning and Inference 45 (1995) 133 141

About each term in the sum above, observe that St.,,. ,~EX, uN.(x)]((O, b]) [n,'`

=E

1]

1 {Xl > u t.,~- 1J(x')} 61/t,,'`_ 11((0, b])

i=1

= St,,'` ,1 [X, ut,,~ 11(x')((0, b]),

(2.4)

with aN.

an.

bN.-- bt.t'`_,j

aInt'`- 11

a[nt'`- 11

aN n

X'-- - - X + - -

Since P(M, <~u,(x)). ~o H(x) = P(S (*)((0, 1]) = 0) and {X,} has extremal index 0 > 0 then P(A/ln<~u.(X))n-~ooH1/°(x ) (Alpuim, 1988). Hence, if we define F ( x ) =

I-IT ~ P~/r(xi~x), then F"(u.(x),~oH1/°(x). By L e m m a 2.2, it follows that lim at'''` IJ=A~_I 1,

lim bt"''` a - b "

.--,~

n~

an

B,'` 1.

aim'`_ d

For a given 6 > 0, if we choose t{s sufficiently close then, using the same arguments as Galambos, we conclude from (2.3) and (2.4) that P(SN. [X, uN.(x)] ((0, b]) ~
<~

P(St, t'` lJ X, ut,t'` ,j ~

((O,b])~s, t k - l < N ~ t k ,

B)+4&

k=l

which, by L e m m a 2.1, converges to P(S (~+ ~)/(1- ~)((0, b]) <~s) P(to < N <~tin, B) + 4e. On the other hand,

P(SN. [X, uN.(x)] ((0, b]) ~1

P(St"t'`l X'ut't'`J 1 + 6

((O,b])<~s, t k _ ~ < N ~ t k , B)-4e,

k=l

which converges to P(S (x- 0)/(1+°)((0, b]) ~
Xi--bN. aN.,

Xi--b. a. an

(X, an

aN n

b.-bN. aN.

A )X,

aN

,25,

H. Ferreira/ Journal of Statistical Plannin9and Inference45 (1995) 133-141

139

and, by L e m m a 2.2, an/as.-AN and the fourth term converge to zero in probability. About the other factor in the third term, for 6 > 0 , n sufficiently large and / > 0 arbitrary chosen, we can write P

E l{(x~_b.)/a.>_t}<~S ~ P i=1

I{(Xi_bN.)/aN>--I(6+AN)+O--BN}~S

"Jt-~

i

<~P

i=

l{(Xi--bN,)/aN>-l(6+a)+6-b} ~S,

to
\i=1

where a > 0 and b are constants defined by to, tm and the parameters of H. Hence, for l sufficiently large, limsu p P (SN. [X, u , ( - / ) ] ((0, b] ) ~
(2.6)

On the other hand, by the first equality in (2.5), P

l((x,-b.)/a.>-~} > 0

~
i=

+2e, i=l

where a ' > 0 and b' are constants. Hence, for l sufficiently large and 6 0) ~< 1 - H(l(--6 + a')-- 6 - b t )-~ 3e. F r o m (2.5)-(2.7) it follows, for each k, that

P(SN. [X, uN.(x)] ((0, b]) <~s, Dk) <~P(SN. [X, uN. (x)] ((0, b]) <~s, Dk, SN. [X, un(-- 1)] ((0, b]) > s, SN.[X, un(l)]((O,b])=O, an--AN <~, b , - b u . aN. aN.

Bu < 6 ) + 5 e

<~P(SN. [X, un(A~ a(x + 26 -- BN))] ((0, b]) <<.s, Dk) + 5e and

P(SN.[X, uN.(x)] ((0, b])~< s, Dk) >~P(SN. [X, un(A~ 1(x -- 26 -- BN))-] ((0, b] ) <<.s, Dk)-- 5e.

(2.7)

140

H. Ferreira/Journal of Statistical Planning and Inferen(e 45 (1995) 133-141

From these inequalities and (i) we obtain lijnoo P(SN.[ X, u.(A~ l(x - BN))] ((0, b])~< s, Dk) = lim P(SN. [X, uN°(x)] ((0, b]) <~s, Dk) =P(S[v(x),lr]((O,b])<~s)P(Ok),

k= 1,...,m.

(2.8)

Let Zk be fixed in (tk-l,tk], k = l ..... m. By Lemma 2.2 we conclude that H(Azkx + Bzk) = HZk(x) and therefore z(Az~x + B~) = ZkZ(X). If we substitute in (2.8) x by x'=A~kx+B~k, if follows that lirn P(SN. [X, u,(A# '(x' -- BN))] ((0, b]) <. s, Dk) = P(S [v(x'), 7z] ((0, b]) ~
m

and lim ~ f P (Ss. [ X, u. (x)] ((0, b] ) <~s, Dk) >~P(S [ZRV((1 -- 6)X

-- 6),

7Z] ((0,

b]) ~ s)P(Dk) E

>~P(S [ZkV(X), ~] ((0, b] ) <~s)P(Dk)----

m"

Then, for large n, I P(Su.EX, u.(x)] ((0, b])<~s)- ~, P(SEzkv(x), lr] ((0, b])~< s)P(D,)I< 2~ k=l

H. Ferreira/Journal of Statistical Planning and Inference 45 (1995) 133-141

141

and

IP(SN.[X,u.(x)]((O,b])<~s)-

P(S[zv(x),rc]((O,b])<~s)dP(N<~z)l<38.

Because 8 is arbitrary, taking limits as n ~ oo, (ii) follows.

[]

Acknowledgements I am grateful to my supervisor, Professor Maria Ivette Gomes, for many helpful suggestions and her encouragement. I would also like to thank the referee for his detailed comments and suggestions which helped considerably in improving the final form of this paper.

References Alpuim, M.T. (1988). Contribui¢6es h teoria de extremos em sucess6es dependentes. Ph.D. thesis, D.E.I.O.C., Univ. of Lisboa. Barakat, H.M. and M.A. El-Shandidy (1990). On the limit distribution of the extremes of a random number of independent random variables. J. Statist. Plann. Inference 26, 353-361. Ferreira, H. (1993). Multivariate extreme values in in T-periodic random sequences under mild oscillation restrictions. Stochastic Process and their Applications 49, 111-125. Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. J. Wiley, New York. Hsing, T. J. H/isler and M.R. Leadbetter (1988). On the exceedance point process for stationary sequence. Probab. Theory Related Fields 78, 97-112. Leadbetter, M.R. (1974). On extreme values in stationary sequences. Z. Wahrsch. verw. Gebiete 28, 289 303. Leadbetter, M.R. and S. Nandagopalan (1989). On exceedance point processes for stationary sequences under mild oscillation restrictions. In: H/isler and R.-D. Reiss, Eds., Extreme Value Theory. Springer, Berlin, 69-80. Nandagopalan, S. (1990). Multivariate extremes and estimation of the extremal index. Ph.D. thesis, Univ. of North Carolina, Chapel Hill, NC.