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Limit theorems for the ratio of weak records
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 1454–1464 www.elsevier.com/locate/stapro
Limit theorems for the ratio of weak records A. Dembin´skaa, A. Stepanovb, a
Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland b Department of Mathematics, Kaliningrad State Technical University, Sovietsky Prospect 1, Kaliningrad 236000 Russia Received 11 October 2005; received in revised form 24 February 2006; accepted 1 March 2006 Available online 18 April 2006
Abstract Let X 1 ; X 2 ; . . . be independent identically distributed (iid) random variables (rvs) with support on positive integers and ðW n ; nX1Þ denote the corresponding sequence of weak record values. In the present paper we derive weak and strong limit theorems for the ratio of weak records W nþm =W n ðn ! 1Þ. We also discuss the asymptotic behavior of a rv xn which counts the number of weak record values that are equal to n. r 2006 Elsevier B.V. All rights reserved. MSC: 60G70; 62G30 Keywords: Weak records; Limit theorems; Slowly, regularly and rapidly varying functions; Regularly varying sequences
1. Introduction The notion of weak records was introduced by Vervaat (1973) as a modification of records for discrete distributions. Let X 1 ; X 2 ; . . . be a sequence of independent identically distributed (iid) random variables (rvs) with support f1; 2; . . . ; Ng, Np1, distribution function F ðxÞ ¼ PðX 1 pxÞ, survival function qðxÞ ¼ qx ¼ PðX 1 XxÞ and probability mass function (pmf) px ¼ PðX 1 ¼ xÞ. The sequences of weak record times ðLn ; nX1Þ and weak record values ðW n ; nX1Þ are defined as follows: L1 ¼ 1; Lnþ1 ¼ minfj : j4Ln ; X j XX Ln g, W n ¼ X Ln ; nX1.
ð1Þ
If in (1) the sign X is substituted with the sign 4 then instead of the sequence of weak record values W n we get the sequence of record values X ðnÞ. A considerable amount of work has been done on weak record statistics. Stepanov (1992) studied asymptotic properties of weak records. Stepanov (1993), Aliev (1998, 1999), Lo´pezBla´zquez and Weso"owski (2001), Weso"owski and Ahsanullah (2001), Weso"owski and Lo´pez-Bla´zquez (2004), Dembin´ska and Lo´pez-Bla´zquez (2005), Danielak and Dembin´ska (2006) presented characterizations of discrete distributions by properties of weak record values. Stepanov et al. (2003) obtained exact distributions of weak record values in the case when the support has finite number of atoms. The Fisher Corresponding author.
E-mail addresses: [email protected] (A. Dembin´ska), [email protected] (A. Stepanov). 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.03.004
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information contained in weak record values was also discussed in that paper. Some topics related to weak records are also presented in the books of Arnold et al. (1998) and Nevzorov (2001). The joint pmf of the first n weak records can be easily derived as PðW 1 ¼ k1 ; . . . ; W n ¼ kn Þ ¼ pkn
n1 Y pk
i
q i¼1 ki
,
(2)
for any 1pk1 p pkn pN (if N ¼ 1 the last inequality is, obviously, sharp). It immediately follows that the sequence ðW n ; nX1Þ forms a Markov chain and the conditional distribution of two weak record values has the form pj (3) PðW nþm ¼ jjW n ¼ iÞ ¼ Rm ði; jÞ; 1pipjpN, qi P P where R1 ðl; tÞ ¼ 1 and Rm ðl; tÞ ¼ tl1 ¼l pl 1 =ql 1 tlm1 ¼l m2 pl m1 =ql m1 , mX2. When dealing with weak records it is convenient to define rvs xi , i ¼ 1; 2; . . . as follows: xi ¼ k
if there are exactly k weak record values that are equal to i.
The proof of the following result can be found in Stepanov (1992). Lemma 1. The rvs xi , i ¼ 1; 2; . . . are independent and qiþ1 qiþ1 k Pðxi ¼ kÞ ¼ 1 ; k ¼ 0; 1; . . . ; i ¼ 1; 2; . . . N 1, qi qi
(4)
where PðxN ¼ 1Þ ¼ 1 if No1 and then PðxNþj ¼ 0Þ ¼ 1, jX1. Lemma 1 is useful when one works with weak records because it allows to pass from dependent weak record values to independent rvs xi due to the following representation. Representation 1. fW n 4mg ¼ fx1 þ x2 þ þ xm ong
for nX1; mX1.
In this paper we study asymptotic behavior of the rvs Z n;m ¼ W nþm =W n (mX1; n ! 1) and xn (n ! 1). We will see that this behavior depends on limiting properties of the parent distribution tail F ¼ 1 F . Let N ¼ 1 and consider the following limit lim bðn; xÞ ¼ bðxÞ,
n!1
(5)
where bðn; xÞ ¼ F ðxnÞ=F ðnÞ and n ¼ 1; 2; . . . : From the theory of regular variation it is known that if the limit in (5) exists for any x40 (possibly infinite if x 2 ð0; 1Þ), then only three cases are possible. Here, we will say that 1. F is a rapidly varying function if bðxÞ ¼ 0ðx41Þ and bðxÞ ¼ 1ð0oxo1Þ, 2. F is regularly varying with index r if bðxÞ ¼ xr ðx40; ro0Þ, 3. F is slowly varying if bðxÞ ¼ 1ðx40Þ. For details see Bingham et al. (1987, p. 52). It needs to be mentioned here that the limit in (5) cannot exist at all. This, for example, happens in the case when the distribution tail is given by 1 þ ð1 cosð2p log nÞÞ=a ; n ¼ 1; 2; . . . , n where a42p. The paper is organized as follows. In Section 2 we recall some concepts and results from the theory of regular variation and present several lemmas concerning distributional properties of the rv W n . Then we turn to our main object and in Sections 3 and 4 derive weak and strong limit theorems for the ratio of weak F ðnÞ ¼
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record values Z n;m as ðn ! 1Þ. In Section 5 we study asymptotic behavior of rvs xn , n ! 1. Finally in Section 6 we discuss several examples. d
In the remainder of the paper we assume that N ¼ 1. Moreover, by ! we denote convergence in law and P
by ! convergence in probability. 2. Preliminary results As mentioned in Introduction asymptotic behavior of the rvs Z n;m and xn as n ! 1 is connected with limiting properties of the parent distribution tail F . To analyze these properties we will need a number of results from the theory of regular variation. Therefore we begin with recalling some basic facts from this theory. Definition 1. A measurable function f : Rþ ! Rþ is regularly varying with index r 2 R if for any x40 lim
t!1
f ðtxÞ ¼ xr . f ðtÞ
(6)
If r ¼ 0 we call f slowly varying. If r ¼ 1 we call f rapidly varying. Note that according to Definition 1 a slowly varying function is also regularly varying. In this paper, to avoid confusion, we will call f slowly varying if (6) holds with r ¼ 0 and regularly varying if (6) holds with r 2 R f0g. Definition 2. A sequence ðcn ; nX1Þ of positive numbers is regularly varying with index r 2 R if for any x40 c½xn lim ¼ xr , n!1 cn where ½ stands for integer part. Lemma 2. If ðcn Þ is regularly varying, cn1 =cn ! 1 as n ! 1. t!1
Note that to establish (6) with r ¼ 1, only f ðtxÞ=f ðtÞ ! 0 for x41 has to be proved. The following lemma shows that we can weaken (6) also when r 2 R. Lemma 3 (Characterization theorem). If f : Rþ ! Rþ is measurable and for any x 2 A lim
t!1
f ðtxÞ ¼ gðxÞ 2 ð0; 1Þ, f ðtÞ
where A is some set of positive measure, then there exist a real number r such that (6) holds. Under additional assumption that f is monotone we can consider the limit in (6) only for integer n rather than real t and only for two suitable chosen values of x’s. Lemma 4. A monotone function f : Rþ ! Rþ varies regularly (or slowly in the case of r ¼ 0) if there exist positive integers m1 , m2 with ðlog m1 Þ=ðlog m2 Þ irrational such that f ðmnÞ ! mr f ðnÞ
as n integer ! 1
for m ¼ m1 ; m2 . Proofs of the above three lemmas can be found in Bingham et al. (1987, p. 52, 17 and 56, respectively). For more details on regularly, slowly and rapidly varying functions the reader can also consult Resnick (1987) and Haan de (1970). The next lemma considers similarity between limiting properties of the survival function and the corresponding tail. Lemma 5. The tail of a distribution function F is regularly varying with index r or slowly varying iff the corresponding survival function q has the same property.
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Proof. Assume that F is a slowly or regularly varying function with index r. Then by representation theorem for regularly varying functions (see Bingham et al., 1987, p. 12, 21), we have Z x dðuÞ du ; xXa, F ðxÞ ¼ cðxÞ exp u a for some a40 where cðÞ and dðÞ are measurable functions such that cðxÞ ! c 2 ð0; 1Þ and dðxÞ ! r as x ! 1. Fix 40. Observe that some T exists such that for all tXT and uXt 1 the inequality r pdðuÞpr þ holds. Thus, for x41 and large t, we get Z xt1 qðxtÞ F ðxt 1Þ cðxt 1Þ dðuÞ p exp du ¼ qðtÞ cðtÞ u F ðtÞ t cðxt 1Þ xt 1 t!1 rþ p exp ðr þ Þ log ! x , cðtÞ t and similarly qðxtÞ F ðxtÞ t!1 r X ! x . qðtÞ F ðt 1Þ Letting ! 0 we obtain for xX maxfa; 1g that lim
t!1
qðxtÞ ¼ xr , qðtÞ
which, by Lemma 3, means that q is slowly (if r ¼ 0) or regularly varying (if ro0). Thus slow or regular variation of F implies that of q. The converse can be shown in a similar way. & The following three lemmas concern distributional properties of weak record values. Lemma 6. For any m ¼ 1; 2; . . . ; i ¼ 1; 2; . . . and x4i we have PðW nþm XxjW n ¼ iÞ ¼
m qx X Rj ði; dxe 1Þ, qi j¼1
where dye means the smallest natural number greater or equal y. Proof. Formula (3) implies that PðW nþm XxjW n ¼ iÞ ¼
X
PðW nþm ¼ jjW n ¼ iÞ ¼
jXx
¼
j j X pj X X pl 1 q q jXx i l ¼i l 1 ¼l l 1
m1
m2
X pj Rm ði; jÞ q jXx i
pl m1 . ql m1
Changing the order of summation (we place the sum over j as the last one) ends the proof. Lemma 7. For any i ¼ 1; 2; . . . 1 X n¼1
PðW n ¼ iÞ ¼
pi . qiþ1
&
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Proof. It follows from Representation 1 and Lemma 1 that for any i ¼ 2; 3; . . . 1 X
PðW n ¼ iÞ ¼
n¼1
¼ ¼
1 X n1 X
Pðx1 þ þ xi1 ¼ k; xi Xn kÞ
n¼1 k¼0 1 X
1 X
Pðx1 þ þ xi1 ¼ kÞ
k¼0 1 X
Pðxi Xn kÞ
n¼kþ1
Pðxi XmÞ ¼ Eðxi Þ ¼
m¼1
pi . qiþ1
However for i ¼ 1 we get 1 X
PðW n ¼ 1Þ ¼
n¼1
1 X
Pðx1 XnÞ ¼
n¼1
Corollary 1. Since
P1 P1 n¼1
i¼1 PðW n
p1 : q2
¼ iÞ ¼
& P1
n¼1 1
¼ 1, Lemma 7 implies that
P1
i¼1 pi =qiþ1
¼ 1:
Lemma 8. If N ¼ 1, then for any choice of discrete F n!1
W n ! 1 a:s.
P P1 Pm Proof. Since for any fixed mX1, 1 n¼1 PðW n pmÞ ¼ n¼1 i¼1 PðW n ¼ iÞ, Lemma 7, upon interchanging the order of summation, implies that 1 m X X pi PðW n pmÞ ¼ o1, qiþ1 n¼1 i¼1 which readily means that W n tends to infinity with probability one.
&
3. Weak limit results for the ratio of weak records Assuming that the support is unbounded from the right and lim
n!1
F ðnxÞ ¼ xr ; x41, F ðnÞ
(7)
where 1prp0, Shorrock (1972) proved that X ðn þ 1Þ Xx ¼ xr ; xX1 lim P n!1 X ðnÞ and the rvs X ðn þ 1Þ=X ðnÞ; X ðn þ 2Þ=X ðn þ 1Þ; . . . ; X ðn þ kÞ=X ðn þ k 1Þ are asymptotically independent for any kX2. Repeating similar arguments one can show that an analogous result holds for weak records. Theorem 1. Let F be a distribution tail such that (7) holds with 1prp0. Then lim P Z n;1 Xx ¼ xr ; xX1 n!1
and the rvs Z n;1 ; Z nþ1;1 ; . . . ; Z nþk;1 are asymptotically independent for any k ¼ 1; 2; . . . : Proof. By Lemma 4, assumption (7) with r 2 ð1; 0 implies that F varies slowly (if r ¼ 0) or regularly (if ro0) and by Lemma 5, q has the same property. If r ¼ 1 then (7) gives 0p lim
n!1
qðxnÞ F ðxðn 1ÞÞ p lim ¼ 0; n!1 F ðn 1Þ qðnÞ
x41.
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Thus we have proved that (7) with 1prp0 implies q lim nx ¼ xr ; x41. n!1 qn
(8)
Now using (3) we obtain that for any xX1 and y 2 f1; 2; . . .g qxy . PðW nþ1 XxyjW n ¼ yÞ ¼ qy Hence
W nþ1 qðxW n Þ PðZ n;1 XxÞ ¼ E P XxjW n ¼E . qðW n Þ Wn
(9)
n!1
By Lemma 8, W n ! 1 a.s. since N ¼ 1. Therefore by (8) lim
n!1
qðxW n Þ ¼ xr qðW n Þ
a:s.
(10)
and by (9) and the bounded convergence theorem lim PðZ n;1 XxÞ ¼ xr ;
n!1
xX1.
(11)
Now we will show that the rvs Zn;1 ; Z nþ1;1 ; . . . ; Z nþk;1 are asymptotically independent for any kX1, that is for any x0 X1; x1 X1; . . . ; xk X1 lim PðZ n;1 Xx0 ; Z nþ1;1 Xx1 ; . . . ; Z nþk;1 Xxk Þ ¼ xr0 xr1 xrk .
(12)
n!1
We will proceed by induction in k. Assuming (12) holds for some k we will prove it for k þ 1. PðZ n;1 Xx0 ; . . . ; Z nþkþ1;1 Xxkþ1 Þ W nþ1 W nþkþ1 W nþkþ2 Xx0 ; . . . ; Xxk ; Xxkþ1 jW n ; . . . ; W nþkþ1 ¼E P Wn W nþk W nþkþ1 W nþ1 W nþkþ1 W nþkþ2 ¼E I Xx0 ; . . . ; Xxk P Xxkþ1 jW n ; . . . ; W nþkþ1 . Wn W nþk W nþkþ1
ð13Þ
Since ðW n ; nX1Þ is a Markov chain, (13) and (3) show that PðZ n;1 Xx0 ; . . . ; Z nþkþ1;1 Xxkþ1 Þ qðxkþ1 W nþkþ1 Þ ¼ E IðZ n;1 Xx0 ; . . . ; Z nþk;1 Xxk Þ qðW nþkþ1 Þ qðxkþ1 W nþkþ1 Þ xrkþ1 ¼ E IðZ n;1 Xx0 ; . . . ; Z nþk;1 Xxk Þ qðW nþkþ1 Þ n!1
þ xrkþ1 PðZ n;1 Xx0 ; . . . ; Z nþk;1 Xxk Þ ! 0 þ xrkþ1 xr0 xrk by (10) and the bounded convergence theorem. Note that (12) holds for k ¼ 0 by (11).
&
Theorem 1 implies the following results. Corollary 2. Let F be a rapidly varying function. Then for any mX1 P
Z n;m ! 1
as n ! 1. P
Proof. Since F is a rapidly varying function, Theorem 1 with r ¼ 1 implies that Z n;1 ! 1. Now notice that P
Z n;m ¼ Z n;1 Z nþ1;1 Z nþm1;1 ! 1 as a product of m rvs which converge in probability to 1. & The result can be commented in the following manner. The rapidly varying tail F is ‘‘thin’’. New weak record value ‘‘jumps’’ near the previous one. So, their ration tends to 1 in probability. Now we consider slowly varying tails.
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Corollary 3. Let F be a slowly varying function. Then for any mX1 P
Zn;m ! 1
as n ! 1. P
Proof. Theorem 1with r ¼ 0 implies that Z n;1 ! 1. Noticing that Zn;m XZ n;1 , we finish the proof.
&
We can comment the above result in the following way. Slowly varying tails F are ‘‘thick’’. That is why, a new weak record value jumps far ahead of the previous one, and their ratio tends to infinity. Corollary 4. Let F be a regularly varying function with index ro0. Then for any mX1 1 þ ðr log xÞ þ þ ðr log xÞm1 =ðm 1Þ! ; xX1. (14) n!1 xr Proof. We prove it by induction in m. For m ¼ 1 the result follows from Theorem 1. Now suppose that (14) holds for m, we will prove it for m þ 1. By the induction assumption and Theorem 1 lim PðZ n;m XxÞ ¼
d
Znþm;1 ! Y
where F Y ðxÞ ¼ PðY 4xÞ ¼
1 ; xr
xX1,
1 þ ðr log xÞ þ þ ðr log xÞm1 =ðm 1Þ! ; xr and the rvs Y and Z are independent. Hence d
Zn;m ! Z
where F Z ðxÞ ¼ PðZ4xÞ ¼
xX1
d
Zn;mþ1 ¼ Zn;m Z nþm;1 ! Z Y . Moreover
Z 1 Z x r x x PðZ Y XxÞ ¼ E P Y X dF Z ðtÞ þ dF Z ðtÞ. ¼ Z t 1 x
Integrating the first integral by parts and using the induction assumption, we get the desired formula.
&
4. Strong limit theorems for the ratio of weak records Imposing additional conditions on the distribution tail F we will transform Corollaries 2 and 3 into strong limit laws. Theorem 2. Let some d40 exists such that 1 X
bðn 1; xÞ
n¼1
pn o1; qnþ1
x 2 ð1; 1 þ dÞ.
Then for all mX1 Zn;m ! 1 a:s:
as n ! 1.
Proof. First using Borel–Cantelli Lemma we show that for any jX0 n!1
Znþj;1 ! 1 a:s. Indeed Lemmas 6 and 7 and (15) imply that for K large enough 1 1 X X 1 1 P jZ nþj;1 1jX P W nþjþ1 X 1 þ ¼ W nþj K K n¼1 n¼1 1 X 1 X 1 ¼ P W nþjþ1 X 1 þ ijW nþj ¼ i PðW nþj ¼ iÞ K n¼1 i¼1 1 q 1 X X F 1 þ 1=K ði 1Þ pi ðð1þ1=K ÞiÞ pi p p o1. qiþ1 qiþ1 qi F ði 1Þ i¼1 i¼1
(15)
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n!1
Now notice that Z n;m ¼ Zn;1 Z nþ1;1 Znþm1;1 ! 1 a.s. as a product of m rvs which converge a.s. to 1. & Remark 1. By Corollary 1 if for any x 2 ð1; 1 þ dÞ the limit limn!1 bðn; xÞ exists then condition (15) implies that F is a rapidly varying function. Indeed, for real positive t 0p
½t!1 F ðtxÞ F ð½txÞ p ¼ bð½t; xÞ ! 0; F ðtÞ F ð½tÞ
x 2 ð1; 1 þ dÞ.
(16)
Moreover, since F ðtyÞ=F ðtÞpF ðtxÞ=F ðtÞ for any y4x, (16) shows that lim
t!1
F ðtyÞ ¼ 0; F ðtÞ
yX1.
Remark 2. Since pn =qn o1 Theorem 2 will continue to hold if we replace condition (15) with 1 X qnx o1; q n¼1 nþ1
x 2 ð1; 1 þ dÞ.
(17)
Theorem 3. If mX1 and K exist such that for all x4K the inequality " # 1 m X qnx X pn 1 Rj ðn; dnxe 1Þ o1 qn j¼1 qnþ1 n¼1
(18)
holds then for all iXm, n!1
Z n;i ! 1 a:s. Proof. Using Lemmas 6 and 7 we get for x4KX1 1 X n¼1
PðZ n;m oxÞ ¼
1 X 1 X
PðW nþm oxijW n ¼ iÞPðW n ¼ iÞ
n¼1 i¼1
" 1 X 1 X
# m qix X ¼ 1 Rj ði; dixe 1Þ PðW n ¼ iÞ qi j¼1 n¼1 i¼1 " # 1 m X qix X pi ¼ 1 Rj ði; dixe 1Þ o1. q q i j¼1 iþ1 i¼1
Now the theorem immediately follows from Borel–Cantelli Lemma.
&
Remark 3. Observe that for any mX1 q q 1 nx T mþ1 ðn; xnÞo1 nx T m ðn; xnÞ, qn qn Pm where T m ðn; xÞ ¼ j¼1 Rj ðn; dxe 1Þ. The last inequality implies that if the series in (18) converges for some m ðmX1Þ, then it also converges for m þ 1. Remark 4. By Corollary 1 and Lemma 4, if qnx =qn converges as n ! 1 then condition (18) with m ¼ 1 yields that q is a slowly varying function. This by Lemma 5, shows that also F is slowly varying. Moreover if we assume that F is slowly varying then by Lemma 5, q is also slowly varying and by Lemma 2, limn!1 qnþ1 =qn ¼ 1. Hence pn =qn ppn =qnþ1 p2ðpn =qn Þ for all large n. Consequently the sum in (18) converges if and only if " # 1 m X qnx X p 1 Rj ðn; dnxe 1Þ n o1 (19) q qn n j¼1 n¼1 which implies the following remark.
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Remark 5. Theorem 3 will continue to hold if we assume that F is slowly varying and replace condition (18) by (19). P Remark 6. Since m j¼1 Rj ðn; dnxe 1Þ41 the conclusion of Theorem 3 will hold for any mX1 if we replace condition (18) with 1 X qnx pn 1 o1. (20) qn qnþ1 n¼1 5. Asymptotic behavior of the rvs xn As we already know, the rvs xn are geometrically distributed with parameters qnþ1 =qn . This immediately implies the following proposition. Proposition 1. P
1. xn ! 1 if qnþ1 =qn ! 0 as n ! 1, d
2. xn ! x if qnþ1 =qn ! q 2 ð0; 1Þ as n ! 1 and the rv x is such that Pðx ¼ iÞ ¼ ð1 qÞqi , i ¼ 0; 1; . . . , P
3. xn ! 0 if qnþ1 =qn ! 1 as n ! 1. P
Note that Lemmas 2, 5 and Proposition 1.3 show that if F is slowly or regularly varying, then xn ! 0 as n ! 1. Further the condition of Proposition 1.1 implies that F is a rapidly varying function, because 8x41 9N 8n4N ;
0p
F ðnxÞ qnþ2 n!1 ! 0. p qnþ1 F ðnÞ
Yet the converse, in general, is not true, for example for the geometric distribution with pmf pn ¼ ð1 qÞqn1 , n ¼ 1; 2; . . . lim
t!1
F ðtxÞ q ¼ lim q½tx½t ¼ 0 ðx41Þ but lim nþ1 ¼ qa0. t!1 n!1 qn F ðtÞ P
Therefore it is not enough to assume that F is a rapidly varying tail if we want to conclude that xn ! 1. Putting an additional condition that qnþ1 =qn tends to 0 quickly enough we can transform Proposition 1.1 into a strong limit law. P Proposition 2. Suppose 1 n¼1 qnþ1 =qn o1. Then xn ! 1 a.s. Proof. It follows from (4) that 1 X
Pðxn pkÞ ¼
n¼1
1 1 X X
1 ðpn =qn Þkþ1 pðk þ 1Þ qnþ1 =qn . n¼1
n¼1
The result now follows from Borel–Cantelli Lemma.
&
Proposition 3. If for some k ¼ 2; 3; . . . 1 X ðpn =qn Þk o1, n¼1
then for the same k P lim xn ok ¼ 1. n!1
(21)
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Proof. Indeed, formula (4) entails 1 X
Pðxn XkÞ ¼
n¼1
1 X
ðpn =qn Þk .
(22)
n¼1
If for some kX2 (this is impossible for k ¼ 1) the series in the right-hand side of (22) converges then Pfxn Xk i:o:g ¼ 0 and the statement of Proposition 3 holds. & Proposition 4. Let qnþ1 =qn ! 1 as n ! 1. Then the sequence of rvs ðxn ; nX1Þ converges nowhere in respect to a.s. convergence. Proof. Assume that the conclusion is false. Then by Proposition 1.3, xn ! 0 a.s. which is in contradiction with Lemma 8. & Proposition 4 and Lemma 2 imply the following result. Corollary 5. Let F be slowly or regularly varying tail. Then the sequence ðxn ; nX1Þ converges nowhere in respect to a.s. convergence. 6. Examples 6.1. First let us consider the ‘‘friendliest’’ geometric distribution with pmf pn ¼ ð1 qÞqn1 , n ¼ 1; 2; . . . : Then F is a rapidly varying tail and by Corollary 2 P
(23) Z n;m ! 1 for mX1 and n ! 1. P P1 nðx1Þ o1, Theorem 2 shows that the convergence Moreover, since n¼1 bðn 1; xÞðpn =qnþ1 Þppqx1 1 n¼1 q in (23) is not only in probability but also with probability 1. Further the rvs xn ; nX1 are iid with pmf Pðxn ¼ iÞ ¼ ð1 qÞqi , i ¼ 0; 1; . . . . 6.2. Let us consider a shifted Poisson distribution with pmf pn ¼ el ðln1 =ðn 1Þ!Þ, n ¼ 1; 2; . . . ; where l40. Then F is also a rapidly varying tail. Indeed, for lX1 and x41 P½nx pi F ðnxÞ 1 þ l=n þ 1 þ þ l½nxn =ðn þ 1Þðn þ 2Þ . . . ð½nx 1Þ ¼1 0p ¼ 1 Pi¼nþ1 1 F ðnÞ 1 þ l=n þ 1 þ l2 =ðn þ 1Þðn þ 2Þ þ i¼nþ1 pi 1 þ lð½nx n 1=ðn þ 1Þðn þ 2Þ . . . ð½nx 1ÞÞ n!1 p1 ! 0 ð1 l=n þ 1Þ1 and repeating the same arguments as used in Remark 1, we obtain that limt!1 F ðtxÞ=F ðtÞ ¼ 0 (x41 and t real). If l 2 ð0; 1Þ we proceed similarly. Consequently by Corollary 2 P
Z n;m ! 1
for mX1 and n ! 1.
Furthermore, 0p
qnþ1 1 l n!1 p ! 0, ¼1 2 qn 1 þ l=n þ l =nðn þ 1Þ þ n P
which by Proposition 1.1 implies that xn ! 1. On the other hand qnþ1 =qn X1 ð1 þ l=nÞ1 ¼ l=n þ l and the hypothesis of Proposition 2 does not hold and we cannot justify whether xn ! 1 a.s. or not. 6.3. Assume that qn ¼ nn ; n ¼ 1; 2; . . . . Then 0p
F ðnxÞ qð½nx þ 1Þ ðn þ 1Þnþ1 n!1 p ¼ ! 0 qðn þ 1Þ F ðnÞ ðnx þ 1Þnxþ1
ðx41Þ
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and using the arguments from Remark 1, we conclude that F is a rapidly varying function. Moreover, since for x41 nþ1 n 2 , qnx =qnþ1 pq½nx =qnþ1 pðn þ 1Þ xn 1 (17) holds and by Remark 2, Z n;m ! 1 a.s. as n ! 1 for any fixed mX1. Observe that n n 1 n!1 ! 0, qnþ1 =qn ¼ nþ1 nþ1 P
which by Proposition 1.1 gives that xn ! 1. However, since ðn=n þ 1Þn 1=n þ 141=eðn þ 1Þ, the hypothesis of Proposition 2 does not hold. 6.4 Let qn ¼ n2n , n ¼ 1; 2; . . . . Then, similarly as in Example 6.3, F is rapidly varying, Z n;m ! 1 a.s. as n ! 1 P
and xn ! 1. Moreover, since qnþ1 =qn ¼ ðn=n þ 1Þ2n 1=ðn þ 1Þ2 p14ð1=ðn þ 1Þ2 Þ, Proposition 2 gives xn ! 1 a.s. P 6.5. Let qn ¼ 1= logðn þ e 1Þ, n ¼ 1; 2; . . . : Then F is a slowly varying function. Hence xn ! 0 and P Z n;m ! 1 ðmX1Þ as n ! 1. Further, since for nX2 q log tð1 þ 1=ðn þ e 1ÞÞ 1 1 p p , 1 nþ1 ¼ logðn þ eÞ ðn þ e 1Þ logðn þ eÞ n log n qn q pn logðx þ eÞ 1 nx , p qn qnþ1 n log2 n n!1
Proposition 3 shows that Pðlimn!1 xn o2Þ ¼ 1 and Remark 6 gives that for any mX1 Zn;m ! 1 a.s. n!1
6.6. Let qn ¼ 1=n; n ¼ 1; 2; . . . . Then for any x41, bðn; xÞ ¼ n þ 1=½nx þ 1 ! 1=x, which by Lemma 4 P
shows that F is regularly varying with index 1. Consequently xn ! 0 as n ! 1 and by Corollary 4 for any fixed mX1 lim PðZ n;m XxÞ ¼
n!1
1 þ log x þ þ ðlog xÞm1 =ðm 1Þ! x
for any xX1.
References Aliev, F.A., 1998. Characterizations of distributions through weak records. J. Appl. Statist. Sci. 8, 13–16. Aliev, F.A., 1999. New characterization of discrete distributions through weak records. Theory Probab. Appl. 44, 756–761 (English translation). Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1998. Records. Wiley, New York. Bingham, N.H., Goldie, C., Teugels, J.F., 1987. Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press, Cambridge. Danielak, K., Dembin´ska, A., 2006. Some characterizations of discrete distributions based on weak records. Statist. Papers, to appear. Dembin´ska, A., Lo´pez-Bla´zquez, F., 2005. A characterization of geometric distribution through kth weak records. Comm. Statist. Theory Methods 34, 2345–2351. Haan de, L., 1970. On Regular Variation and its Applications to the Weak Convergence of Sample Extremes. Mathematicae Centre Tract 32, Amsterdam. Lo´pez-Bla´zquez, F., Weso"owski, J., 2001. Discrete distributions for which the regression of the first record on the second is linear. Test 10, 121–131. Nevzorov, V.B., 2001. Records: Mathematical Theory, Translation of Mathematical Monographs, vol. 194. American Mathematical Society, Providence, RI. Resnick, S.I., 1987. Extreme Values, Regular Variation, and Point Processes. Springer, New York. Shorrock, R.W., 1972. On record values and record times. J. Appl. Probab. 9, 316–326. Stepanov, A.V., 1992. Limit theorems for weak records. Theory Probab. Appl. 37, 570–574 (English translation). Stepanov, A.V., 1993. A characterization theorem for weak records. Theory Probab. Appl. 38, 762–764 (English translation). Stepanov, A.V., Balakrishnan, N., Hofmann, G., 2003. Exact distribution and Fisher information of weak record values. Statist. Probab. Lett. 64, 69–81. Vervaat, W., 1973. Limit theorems for records from discrete distributions. Stochastic Process. Appl. 1, 317–334. Weso"owski, J., Ahsanullah, M., 2001. Linearity of regression for non-adjacent weak records. Statistica Sinica 11 (1), 39–52. Weso"owski, J., Lo´pez-Bla´zquez, F., 2004. Linearity of regression for the past weak and ordinary records. Statistics 38, 457–464.
Statistics & Probability Letters 76 (2006) 1454–1464 www.elsevier.com/locate/stapro
Limit theorems for the ratio of weak records A. Dembin´skaa, A. Stepanovb, a
Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland b Department of Mathematics, Kaliningrad State Technical University, Sovietsky Prospect 1, Kaliningrad 236000 Russia Received 11 October 2005; received in revised form 24 February 2006; accepted 1 March 2006 Available online 18 April 2006
Abstract Let X 1 ; X 2 ; . . . be independent identically distributed (iid) random variables (rvs) with support on positive integers and ðW n ; nX1Þ denote the corresponding sequence of weak record values. In the present paper we derive weak and strong limit theorems for the ratio of weak records W nþm =W n ðn ! 1Þ. We also discuss the asymptotic behavior of a rv xn which counts the number of weak record values that are equal to n. r 2006 Elsevier B.V. All rights reserved. MSC: 60G70; 62G30 Keywords: Weak records; Limit theorems; Slowly, regularly and rapidly varying functions; Regularly varying sequences
1. Introduction The notion of weak records was introduced by Vervaat (1973) as a modification of records for discrete distributions. Let X 1 ; X 2 ; . . . be a sequence of independent identically distributed (iid) random variables (rvs) with support f1; 2; . . . ; Ng, Np1, distribution function F ðxÞ ¼ PðX 1 pxÞ, survival function qðxÞ ¼ qx ¼ PðX 1 XxÞ and probability mass function (pmf) px ¼ PðX 1 ¼ xÞ. The sequences of weak record times ðLn ; nX1Þ and weak record values ðW n ; nX1Þ are defined as follows: L1 ¼ 1; Lnþ1 ¼ minfj : j4Ln ; X j XX Ln g, W n ¼ X Ln ; nX1.
ð1Þ
If in (1) the sign X is substituted with the sign 4 then instead of the sequence of weak record values W n we get the sequence of record values X ðnÞ. A considerable amount of work has been done on weak record statistics. Stepanov (1992) studied asymptotic properties of weak records. Stepanov (1993), Aliev (1998, 1999), Lo´pezBla´zquez and Weso"owski (2001), Weso"owski and Ahsanullah (2001), Weso"owski and Lo´pez-Bla´zquez (2004), Dembin´ska and Lo´pez-Bla´zquez (2005), Danielak and Dembin´ska (2006) presented characterizations of discrete distributions by properties of weak record values. Stepanov et al. (2003) obtained exact distributions of weak record values in the case when the support has finite number of atoms. The Fisher Corresponding author.
E-mail addresses: [email protected] (A. Dembin´ska), [email protected] (A. Stepanov). 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.03.004
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information contained in weak record values was also discussed in that paper. Some topics related to weak records are also presented in the books of Arnold et al. (1998) and Nevzorov (2001). The joint pmf of the first n weak records can be easily derived as PðW 1 ¼ k1 ; . . . ; W n ¼ kn Þ ¼ pkn
n1 Y pk
i
q i¼1 ki
,
(2)
for any 1pk1 p pkn pN (if N ¼ 1 the last inequality is, obviously, sharp). It immediately follows that the sequence ðW n ; nX1Þ forms a Markov chain and the conditional distribution of two weak record values has the form pj (3) PðW nþm ¼ jjW n ¼ iÞ ¼ Rm ði; jÞ; 1pipjpN, qi P P where R1 ðl; tÞ ¼ 1 and Rm ðl; tÞ ¼ tl1 ¼l pl 1 =ql 1 tlm1 ¼l m2 pl m1 =ql m1 , mX2. When dealing with weak records it is convenient to define rvs xi , i ¼ 1; 2; . . . as follows: xi ¼ k
if there are exactly k weak record values that are equal to i.
The proof of the following result can be found in Stepanov (1992). Lemma 1. The rvs xi , i ¼ 1; 2; . . . are independent and qiþ1 qiþ1 k Pðxi ¼ kÞ ¼ 1 ; k ¼ 0; 1; . . . ; i ¼ 1; 2; . . . N 1, qi qi
(4)
where PðxN ¼ 1Þ ¼ 1 if No1 and then PðxNþj ¼ 0Þ ¼ 1, jX1. Lemma 1 is useful when one works with weak records because it allows to pass from dependent weak record values to independent rvs xi due to the following representation. Representation 1. fW n 4mg ¼ fx1 þ x2 þ þ xm ong
for nX1; mX1.
In this paper we study asymptotic behavior of the rvs Z n;m ¼ W nþm =W n (mX1; n ! 1) and xn (n ! 1). We will see that this behavior depends on limiting properties of the parent distribution tail F ¼ 1 F . Let N ¼ 1 and consider the following limit lim bðn; xÞ ¼ bðxÞ,
n!1
(5)
where bðn; xÞ ¼ F ðxnÞ=F ðnÞ and n ¼ 1; 2; . . . : From the theory of regular variation it is known that if the limit in (5) exists for any x40 (possibly infinite if x 2 ð0; 1Þ), then only three cases are possible. Here, we will say that 1. F is a rapidly varying function if bðxÞ ¼ 0ðx41Þ and bðxÞ ¼ 1ð0oxo1Þ, 2. F is regularly varying with index r if bðxÞ ¼ xr ðx40; ro0Þ, 3. F is slowly varying if bðxÞ ¼ 1ðx40Þ. For details see Bingham et al. (1987, p. 52). It needs to be mentioned here that the limit in (5) cannot exist at all. This, for example, happens in the case when the distribution tail is given by 1 þ ð1 cosð2p log nÞÞ=a ; n ¼ 1; 2; . . . , n where a42p. The paper is organized as follows. In Section 2 we recall some concepts and results from the theory of regular variation and present several lemmas concerning distributional properties of the rv W n . Then we turn to our main object and in Sections 3 and 4 derive weak and strong limit theorems for the ratio of weak F ðnÞ ¼
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record values Z n;m as ðn ! 1Þ. In Section 5 we study asymptotic behavior of rvs xn , n ! 1. Finally in Section 6 we discuss several examples. d
In the remainder of the paper we assume that N ¼ 1. Moreover, by ! we denote convergence in law and P
by ! convergence in probability. 2. Preliminary results As mentioned in Introduction asymptotic behavior of the rvs Z n;m and xn as n ! 1 is connected with limiting properties of the parent distribution tail F . To analyze these properties we will need a number of results from the theory of regular variation. Therefore we begin with recalling some basic facts from this theory. Definition 1. A measurable function f : Rþ ! Rþ is regularly varying with index r 2 R if for any x40 lim
t!1
f ðtxÞ ¼ xr . f ðtÞ
(6)
If r ¼ 0 we call f slowly varying. If r ¼ 1 we call f rapidly varying. Note that according to Definition 1 a slowly varying function is also regularly varying. In this paper, to avoid confusion, we will call f slowly varying if (6) holds with r ¼ 0 and regularly varying if (6) holds with r 2 R f0g. Definition 2. A sequence ðcn ; nX1Þ of positive numbers is regularly varying with index r 2 R if for any x40 c½xn lim ¼ xr , n!1 cn where ½ stands for integer part. Lemma 2. If ðcn Þ is regularly varying, cn1 =cn ! 1 as n ! 1. t!1
Note that to establish (6) with r ¼ 1, only f ðtxÞ=f ðtÞ ! 0 for x41 has to be proved. The following lemma shows that we can weaken (6) also when r 2 R. Lemma 3 (Characterization theorem). If f : Rþ ! Rþ is measurable and for any x 2 A lim
t!1
f ðtxÞ ¼ gðxÞ 2 ð0; 1Þ, f ðtÞ
where A is some set of positive measure, then there exist a real number r such that (6) holds. Under additional assumption that f is monotone we can consider the limit in (6) only for integer n rather than real t and only for two suitable chosen values of x’s. Lemma 4. A monotone function f : Rþ ! Rþ varies regularly (or slowly in the case of r ¼ 0) if there exist positive integers m1 , m2 with ðlog m1 Þ=ðlog m2 Þ irrational such that f ðmnÞ ! mr f ðnÞ
as n integer ! 1
for m ¼ m1 ; m2 . Proofs of the above three lemmas can be found in Bingham et al. (1987, p. 52, 17 and 56, respectively). For more details on regularly, slowly and rapidly varying functions the reader can also consult Resnick (1987) and Haan de (1970). The next lemma considers similarity between limiting properties of the survival function and the corresponding tail. Lemma 5. The tail of a distribution function F is regularly varying with index r or slowly varying iff the corresponding survival function q has the same property.
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Proof. Assume that F is a slowly or regularly varying function with index r. Then by representation theorem for regularly varying functions (see Bingham et al., 1987, p. 12, 21), we have Z x dðuÞ du ; xXa, F ðxÞ ¼ cðxÞ exp u a for some a40 where cðÞ and dðÞ are measurable functions such that cðxÞ ! c 2 ð0; 1Þ and dðxÞ ! r as x ! 1. Fix 40. Observe that some T exists such that for all tXT and uXt 1 the inequality r pdðuÞpr þ holds. Thus, for x41 and large t, we get Z xt1 qðxtÞ F ðxt 1Þ cðxt 1Þ dðuÞ p exp du ¼ qðtÞ cðtÞ u F ðtÞ t cðxt 1Þ xt 1 t!1 rþ p exp ðr þ Þ log ! x , cðtÞ t and similarly qðxtÞ F ðxtÞ t!1 r X ! x . qðtÞ F ðt 1Þ Letting ! 0 we obtain for xX maxfa; 1g that lim
t!1
qðxtÞ ¼ xr , qðtÞ
which, by Lemma 3, means that q is slowly (if r ¼ 0) or regularly varying (if ro0). Thus slow or regular variation of F implies that of q. The converse can be shown in a similar way. & The following three lemmas concern distributional properties of weak record values. Lemma 6. For any m ¼ 1; 2; . . . ; i ¼ 1; 2; . . . and x4i we have PðW nþm XxjW n ¼ iÞ ¼
m qx X Rj ði; dxe 1Þ, qi j¼1
where dye means the smallest natural number greater or equal y. Proof. Formula (3) implies that PðW nþm XxjW n ¼ iÞ ¼
X
PðW nþm ¼ jjW n ¼ iÞ ¼
jXx
¼
j j X pj X X pl 1 q q jXx i l ¼i l 1 ¼l l 1
m1
m2
X pj Rm ði; jÞ q jXx i
pl m1 . ql m1
Changing the order of summation (we place the sum over j as the last one) ends the proof. Lemma 7. For any i ¼ 1; 2; . . . 1 X n¼1
PðW n ¼ iÞ ¼
pi . qiþ1
&
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Proof. It follows from Representation 1 and Lemma 1 that for any i ¼ 2; 3; . . . 1 X
PðW n ¼ iÞ ¼
n¼1
¼ ¼
1 X n1 X
Pðx1 þ þ xi1 ¼ k; xi Xn kÞ
n¼1 k¼0 1 X
1 X
Pðx1 þ þ xi1 ¼ kÞ
k¼0 1 X
Pðxi Xn kÞ
n¼kþ1
Pðxi XmÞ ¼ Eðxi Þ ¼
m¼1
pi . qiþ1
However for i ¼ 1 we get 1 X
PðW n ¼ 1Þ ¼
n¼1
1 X
Pðx1 XnÞ ¼
n¼1
Corollary 1. Since
P1 P1 n¼1
i¼1 PðW n
p1 : q2
¼ iÞ ¼
& P1
n¼1 1
¼ 1, Lemma 7 implies that
P1
i¼1 pi =qiþ1
¼ 1:
Lemma 8. If N ¼ 1, then for any choice of discrete F n!1
W n ! 1 a:s.
P P1 Pm Proof. Since for any fixed mX1, 1 n¼1 PðW n pmÞ ¼ n¼1 i¼1 PðW n ¼ iÞ, Lemma 7, upon interchanging the order of summation, implies that 1 m X X pi PðW n pmÞ ¼ o1, qiþ1 n¼1 i¼1 which readily means that W n tends to infinity with probability one.
&
3. Weak limit results for the ratio of weak records Assuming that the support is unbounded from the right and lim
n!1
F ðnxÞ ¼ xr ; x41, F ðnÞ
(7)
where 1prp0, Shorrock (1972) proved that X ðn þ 1Þ Xx ¼ xr ; xX1 lim P n!1 X ðnÞ and the rvs X ðn þ 1Þ=X ðnÞ; X ðn þ 2Þ=X ðn þ 1Þ; . . . ; X ðn þ kÞ=X ðn þ k 1Þ are asymptotically independent for any kX2. Repeating similar arguments one can show that an analogous result holds for weak records. Theorem 1. Let F be a distribution tail such that (7) holds with 1prp0. Then lim P Z n;1 Xx ¼ xr ; xX1 n!1
and the rvs Z n;1 ; Z nþ1;1 ; . . . ; Z nþk;1 are asymptotically independent for any k ¼ 1; 2; . . . : Proof. By Lemma 4, assumption (7) with r 2 ð1; 0 implies that F varies slowly (if r ¼ 0) or regularly (if ro0) and by Lemma 5, q has the same property. If r ¼ 1 then (7) gives 0p lim
n!1
qðxnÞ F ðxðn 1ÞÞ p lim ¼ 0; n!1 F ðn 1Þ qðnÞ
x41.
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Thus we have proved that (7) with 1prp0 implies q lim nx ¼ xr ; x41. n!1 qn
(8)
Now using (3) we obtain that for any xX1 and y 2 f1; 2; . . .g qxy . PðW nþ1 XxyjW n ¼ yÞ ¼ qy Hence
W nþ1 qðxW n Þ PðZ n;1 XxÞ ¼ E P XxjW n ¼E . qðW n Þ Wn
(9)
n!1
By Lemma 8, W n ! 1 a.s. since N ¼ 1. Therefore by (8) lim
n!1
qðxW n Þ ¼ xr qðW n Þ
a:s.
(10)
and by (9) and the bounded convergence theorem lim PðZ n;1 XxÞ ¼ xr ;
n!1
xX1.
(11)
Now we will show that the rvs Zn;1 ; Z nþ1;1 ; . . . ; Z nþk;1 are asymptotically independent for any kX1, that is for any x0 X1; x1 X1; . . . ; xk X1 lim PðZ n;1 Xx0 ; Z nþ1;1 Xx1 ; . . . ; Z nþk;1 Xxk Þ ¼ xr0 xr1 xrk .
(12)
n!1
We will proceed by induction in k. Assuming (12) holds for some k we will prove it for k þ 1. PðZ n;1 Xx0 ; . . . ; Z nþkþ1;1 Xxkþ1 Þ W nþ1 W nþkþ1 W nþkþ2 Xx0 ; . . . ; Xxk ; Xxkþ1 jW n ; . . . ; W nþkþ1 ¼E P Wn W nþk W nþkþ1 W nþ1 W nþkþ1 W nþkþ2 ¼E I Xx0 ; . . . ; Xxk P Xxkþ1 jW n ; . . . ; W nþkþ1 . Wn W nþk W nþkþ1
ð13Þ
Since ðW n ; nX1Þ is a Markov chain, (13) and (3) show that PðZ n;1 Xx0 ; . . . ; Z nþkþ1;1 Xxkþ1 Þ qðxkþ1 W nþkþ1 Þ ¼ E IðZ n;1 Xx0 ; . . . ; Z nþk;1 Xxk Þ qðW nþkþ1 Þ qðxkþ1 W nþkþ1 Þ xrkþ1 ¼ E IðZ n;1 Xx0 ; . . . ; Z nþk;1 Xxk Þ qðW nþkþ1 Þ n!1
þ xrkþ1 PðZ n;1 Xx0 ; . . . ; Z nþk;1 Xxk Þ ! 0 þ xrkþ1 xr0 xrk by (10) and the bounded convergence theorem. Note that (12) holds for k ¼ 0 by (11).
&
Theorem 1 implies the following results. Corollary 2. Let F be a rapidly varying function. Then for any mX1 P
Z n;m ! 1
as n ! 1. P
Proof. Since F is a rapidly varying function, Theorem 1 with r ¼ 1 implies that Z n;1 ! 1. Now notice that P
Z n;m ¼ Z n;1 Z nþ1;1 Z nþm1;1 ! 1 as a product of m rvs which converge in probability to 1. & The result can be commented in the following manner. The rapidly varying tail F is ‘‘thin’’. New weak record value ‘‘jumps’’ near the previous one. So, their ration tends to 1 in probability. Now we consider slowly varying tails.
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Corollary 3. Let F be a slowly varying function. Then for any mX1 P
Zn;m ! 1
as n ! 1. P
Proof. Theorem 1with r ¼ 0 implies that Z n;1 ! 1. Noticing that Zn;m XZ n;1 , we finish the proof.
&
We can comment the above result in the following way. Slowly varying tails F are ‘‘thick’’. That is why, a new weak record value jumps far ahead of the previous one, and their ratio tends to infinity. Corollary 4. Let F be a regularly varying function with index ro0. Then for any mX1 1 þ ðr log xÞ þ þ ðr log xÞm1 =ðm 1Þ! ; xX1. (14) n!1 xr Proof. We prove it by induction in m. For m ¼ 1 the result follows from Theorem 1. Now suppose that (14) holds for m, we will prove it for m þ 1. By the induction assumption and Theorem 1 lim PðZ n;m XxÞ ¼
d
Znþm;1 ! Y
where F Y ðxÞ ¼ PðY 4xÞ ¼
1 ; xr
xX1,
1 þ ðr log xÞ þ þ ðr log xÞm1 =ðm 1Þ! ; xr and the rvs Y and Z are independent. Hence d
Zn;m ! Z
where F Z ðxÞ ¼ PðZ4xÞ ¼
xX1
d
Zn;mþ1 ¼ Zn;m Z nþm;1 ! Z Y . Moreover
Z 1 Z x r x x PðZ Y XxÞ ¼ E P Y X dF Z ðtÞ þ dF Z ðtÞ. ¼ Z t 1 x
Integrating the first integral by parts and using the induction assumption, we get the desired formula.
&
4. Strong limit theorems for the ratio of weak records Imposing additional conditions on the distribution tail F we will transform Corollaries 2 and 3 into strong limit laws. Theorem 2. Let some d40 exists such that 1 X
bðn 1; xÞ
n¼1
pn o1; qnþ1
x 2 ð1; 1 þ dÞ.
Then for all mX1 Zn;m ! 1 a:s:
as n ! 1.
Proof. First using Borel–Cantelli Lemma we show that for any jX0 n!1
Znþj;1 ! 1 a:s. Indeed Lemmas 6 and 7 and (15) imply that for K large enough 1 1 X X 1 1 P jZ nþj;1 1jX P W nþjþ1 X 1 þ ¼ W nþj K K n¼1 n¼1 1 X 1 X 1 ¼ P W nþjþ1 X 1 þ ijW nþj ¼ i PðW nþj ¼ iÞ K n¼1 i¼1 1 q 1 X X F 1 þ 1=K ði 1Þ pi ðð1þ1=K ÞiÞ pi p p o1. qiþ1 qiþ1 qi F ði 1Þ i¼1 i¼1
(15)
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n!1
Now notice that Z n;m ¼ Zn;1 Z nþ1;1 Znþm1;1 ! 1 a.s. as a product of m rvs which converge a.s. to 1. & Remark 1. By Corollary 1 if for any x 2 ð1; 1 þ dÞ the limit limn!1 bðn; xÞ exists then condition (15) implies that F is a rapidly varying function. Indeed, for real positive t 0p
½t!1 F ðtxÞ F ð½txÞ p ¼ bð½t; xÞ ! 0; F ðtÞ F ð½tÞ
x 2 ð1; 1 þ dÞ.
(16)
Moreover, since F ðtyÞ=F ðtÞpF ðtxÞ=F ðtÞ for any y4x, (16) shows that lim
t!1
F ðtyÞ ¼ 0; F ðtÞ
yX1.
Remark 2. Since pn =qn o1 Theorem 2 will continue to hold if we replace condition (15) with 1 X qnx o1; q n¼1 nþ1
x 2 ð1; 1 þ dÞ.
(17)
Theorem 3. If mX1 and K exist such that for all x4K the inequality " # 1 m X qnx X pn 1 Rj ðn; dnxe 1Þ o1 qn j¼1 qnþ1 n¼1
(18)
holds then for all iXm, n!1
Z n;i ! 1 a:s. Proof. Using Lemmas 6 and 7 we get for x4KX1 1 X n¼1
PðZ n;m oxÞ ¼
1 X 1 X
PðW nþm oxijW n ¼ iÞPðW n ¼ iÞ
n¼1 i¼1
" 1 X 1 X
# m qix X ¼ 1 Rj ði; dixe 1Þ PðW n ¼ iÞ qi j¼1 n¼1 i¼1 " # 1 m X qix X pi ¼ 1 Rj ði; dixe 1Þ o1. q q i j¼1 iþ1 i¼1
Now the theorem immediately follows from Borel–Cantelli Lemma.
&
Remark 3. Observe that for any mX1 q q 1 nx T mþ1 ðn; xnÞo1 nx T m ðn; xnÞ, qn qn Pm where T m ðn; xÞ ¼ j¼1 Rj ðn; dxe 1Þ. The last inequality implies that if the series in (18) converges for some m ðmX1Þ, then it also converges for m þ 1. Remark 4. By Corollary 1 and Lemma 4, if qnx =qn converges as n ! 1 then condition (18) with m ¼ 1 yields that q is a slowly varying function. This by Lemma 5, shows that also F is slowly varying. Moreover if we assume that F is slowly varying then by Lemma 5, q is also slowly varying and by Lemma 2, limn!1 qnþ1 =qn ¼ 1. Hence pn =qn ppn =qnþ1 p2ðpn =qn Þ for all large n. Consequently the sum in (18) converges if and only if " # 1 m X qnx X p 1 Rj ðn; dnxe 1Þ n o1 (19) q qn n j¼1 n¼1 which implies the following remark.
ARTICLE IN PRESS A. Dembin´ska, A. Stepanov / Statistics & Probability Letters 76 (2006) 1454–1464
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Remark 5. Theorem 3 will continue to hold if we assume that F is slowly varying and replace condition (18) by (19). P Remark 6. Since m j¼1 Rj ðn; dnxe 1Þ41 the conclusion of Theorem 3 will hold for any mX1 if we replace condition (18) with 1 X qnx pn 1 o1. (20) qn qnþ1 n¼1 5. Asymptotic behavior of the rvs xn As we already know, the rvs xn are geometrically distributed with parameters qnþ1 =qn . This immediately implies the following proposition. Proposition 1. P
1. xn ! 1 if qnþ1 =qn ! 0 as n ! 1, d
2. xn ! x if qnþ1 =qn ! q 2 ð0; 1Þ as n ! 1 and the rv x is such that Pðx ¼ iÞ ¼ ð1 qÞqi , i ¼ 0; 1; . . . , P
3. xn ! 0 if qnþ1 =qn ! 1 as n ! 1. P
Note that Lemmas 2, 5 and Proposition 1.3 show that if F is slowly or regularly varying, then xn ! 0 as n ! 1. Further the condition of Proposition 1.1 implies that F is a rapidly varying function, because 8x41 9N 8n4N ;
0p
F ðnxÞ qnþ2 n!1 ! 0. p qnþ1 F ðnÞ
Yet the converse, in general, is not true, for example for the geometric distribution with pmf pn ¼ ð1 qÞqn1 , n ¼ 1; 2; . . . lim
t!1
F ðtxÞ q ¼ lim q½tx½t ¼ 0 ðx41Þ but lim nþ1 ¼ qa0. t!1 n!1 qn F ðtÞ P
Therefore it is not enough to assume that F is a rapidly varying tail if we want to conclude that xn ! 1. Putting an additional condition that qnþ1 =qn tends to 0 quickly enough we can transform Proposition 1.1 into a strong limit law. P Proposition 2. Suppose 1 n¼1 qnþ1 =qn o1. Then xn ! 1 a.s. Proof. It follows from (4) that 1 X
Pðxn pkÞ ¼
n¼1
1 1 X X
1 ðpn =qn Þkþ1 pðk þ 1Þ qnþ1 =qn . n¼1
n¼1
The result now follows from Borel–Cantelli Lemma.
&
Proposition 3. If for some k ¼ 2; 3; . . . 1 X ðpn =qn Þk o1, n¼1
then for the same k P lim xn ok ¼ 1. n!1
(21)
ARTICLE IN PRESS A. Dembin´ska, A. Stepanov / Statistics & Probability Letters 76 (2006) 1454–1464
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Proof. Indeed, formula (4) entails 1 X
Pðxn XkÞ ¼
n¼1
1 X
ðpn =qn Þk .
(22)
n¼1
If for some kX2 (this is impossible for k ¼ 1) the series in the right-hand side of (22) converges then Pfxn Xk i:o:g ¼ 0 and the statement of Proposition 3 holds. & Proposition 4. Let qnþ1 =qn ! 1 as n ! 1. Then the sequence of rvs ðxn ; nX1Þ converges nowhere in respect to a.s. convergence. Proof. Assume that the conclusion is false. Then by Proposition 1.3, xn ! 0 a.s. which is in contradiction with Lemma 8. & Proposition 4 and Lemma 2 imply the following result. Corollary 5. Let F be slowly or regularly varying tail. Then the sequence ðxn ; nX1Þ converges nowhere in respect to a.s. convergence. 6. Examples 6.1. First let us consider the ‘‘friendliest’’ geometric distribution with pmf pn ¼ ð1 qÞqn1 , n ¼ 1; 2; . . . : Then F is a rapidly varying tail and by Corollary 2 P
(23) Z n;m ! 1 for mX1 and n ! 1. P P1 nðx1Þ o1, Theorem 2 shows that the convergence Moreover, since n¼1 bðn 1; xÞðpn =qnþ1 Þppqx1 1 n¼1 q in (23) is not only in probability but also with probability 1. Further the rvs xn ; nX1 are iid with pmf Pðxn ¼ iÞ ¼ ð1 qÞqi , i ¼ 0; 1; . . . . 6.2. Let us consider a shifted Poisson distribution with pmf pn ¼ el ðln1 =ðn 1Þ!Þ, n ¼ 1; 2; . . . ; where l40. Then F is also a rapidly varying tail. Indeed, for lX1 and x41 P½nx pi F ðnxÞ 1 þ l=n þ 1 þ þ l½nxn =ðn þ 1Þðn þ 2Þ . . . ð½nx 1Þ ¼1 0p ¼ 1 Pi¼nþ1 1 F ðnÞ 1 þ l=n þ 1 þ l2 =ðn þ 1Þðn þ 2Þ þ i¼nþ1 pi 1 þ lð½nx n 1=ðn þ 1Þðn þ 2Þ . . . ð½nx 1ÞÞ n!1 p1 ! 0 ð1 l=n þ 1Þ1 and repeating the same arguments as used in Remark 1, we obtain that limt!1 F ðtxÞ=F ðtÞ ¼ 0 (x41 and t real). If l 2 ð0; 1Þ we proceed similarly. Consequently by Corollary 2 P
Z n;m ! 1
for mX1 and n ! 1.
Furthermore, 0p
qnþ1 1 l n!1 p ! 0, ¼1 2 qn 1 þ l=n þ l =nðn þ 1Þ þ n P
which by Proposition 1.1 implies that xn ! 1. On the other hand qnþ1 =qn X1 ð1 þ l=nÞ1 ¼ l=n þ l and the hypothesis of Proposition 2 does not hold and we cannot justify whether xn ! 1 a.s. or not. 6.3. Assume that qn ¼ nn ; n ¼ 1; 2; . . . . Then 0p
F ðnxÞ qð½nx þ 1Þ ðn þ 1Þnþ1 n!1 p ¼ ! 0 qðn þ 1Þ F ðnÞ ðnx þ 1Þnxþ1
ðx41Þ
ARTICLE IN PRESS A. Dembin´ska, A. Stepanov / Statistics & Probability Letters 76 (2006) 1454–1464
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and using the arguments from Remark 1, we conclude that F is a rapidly varying function. Moreover, since for x41 nþ1 n 2 , qnx =qnþ1 pq½nx =qnþ1 pðn þ 1Þ xn 1 (17) holds and by Remark 2, Z n;m ! 1 a.s. as n ! 1 for any fixed mX1. Observe that n n 1 n!1 ! 0, qnþ1 =qn ¼ nþ1 nþ1 P
which by Proposition 1.1 gives that xn ! 1. However, since ðn=n þ 1Þn 1=n þ 141=eðn þ 1Þ, the hypothesis of Proposition 2 does not hold. 6.4 Let qn ¼ n2n , n ¼ 1; 2; . . . . Then, similarly as in Example 6.3, F is rapidly varying, Z n;m ! 1 a.s. as n ! 1 P
and xn ! 1. Moreover, since qnþ1 =qn ¼ ðn=n þ 1Þ2n 1=ðn þ 1Þ2 p14ð1=ðn þ 1Þ2 Þ, Proposition 2 gives xn ! 1 a.s. P 6.5. Let qn ¼ 1= logðn þ e 1Þ, n ¼ 1; 2; . . . : Then F is a slowly varying function. Hence xn ! 0 and P Z n;m ! 1 ðmX1Þ as n ! 1. Further, since for nX2 q log tð1 þ 1=ðn þ e 1ÞÞ 1 1 p p , 1 nþ1 ¼ logðn þ eÞ ðn þ e 1Þ logðn þ eÞ n log n qn q pn logðx þ eÞ 1 nx , p qn qnþ1 n log2 n n!1
Proposition 3 shows that Pðlimn!1 xn o2Þ ¼ 1 and Remark 6 gives that for any mX1 Zn;m ! 1 a.s. n!1
6.6. Let qn ¼ 1=n; n ¼ 1; 2; . . . . Then for any x41, bðn; xÞ ¼ n þ 1=½nx þ 1 ! 1=x, which by Lemma 4 P
shows that F is regularly varying with index 1. Consequently xn ! 0 as n ! 1 and by Corollary 4 for any fixed mX1 lim PðZ n;m XxÞ ¼
n!1
1 þ log x þ þ ðlog xÞm1 =ðm 1Þ! x
for any xX1.
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