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Asymptotic properties of the ratio of order statistics
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Statistics & Probability Letters 78 (2008) 301–310 www.elsevier.com/locate/stapro
Asymptotic properties of the ratio of order statistics N. Balakrishnana, A. Stepanovb, a
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West,Hamilton, Ont., Canada L8S4K1 b Department of Mathematics, Kaliningrad State Technical University, Sovietsky Prospect 1, Kaliningrad 236000, Russia Received 2 July 2007; received in revised form 2 August 2007; accepted 6 August 2007 Available online 17 August 2007
Abstract Let X 1;n pX 2;n p pX n;n be the order statistics obtained from a sample from a continuous distribution and Zi;j;n ¼ X j;n =X i;n be the ratio of order statistics. In this paper, asymptotic properties of Z i;j;n are discussed. Specifically, we make use of the theory of regular variation to derive weak and strong limit results for Zi;j;n . r 2007 Elsevier B.V. All rights reserved. MSC: 60G70; 62G30 Keywords: Order statistics; Ratio of order statistics; Limit laws; Theory of regular variation
1. Introduction The properties of order statistics have been studied quite extensively in the literature; one may refer to the books by Arnold et al. (1992), Nevzorov (2001), David and Nagaraja (2003), and the references therein. Yet, the asymptotic behaviour of the ratio of order statistics has not been thoroughly investigated. The method based on the theory of regular variation has been used for studying asymptotic properties of the ratio of record values and of weak record values recently by Balakrishnan et al. (2005) and Dembinska and Stepanov (2006), respectively; see also Shorrock (1972) for some early results in this direction. It appears that this method can be extended further and used for examining asymptotic properties of the ratio of order statistics. In this paper, we assume that X 1 ; X 2 ; . . . ; X n are independent and identically distributed random variables with a continuous distribution function F ðxÞ. Let X 1;n pX 2;n p pX n;n be the order statistics based on this sample, and Z i;j;n ¼ X j;n =X i;n ð1piojpnÞ be the ratio of order statistics. For the asymptotic behaviour of the ratio of order statistics, to begin with, we settle on the following aspects. Let i ¼ ½an þ 1, where ½ means the integral part of the number, 0oao1; n ! 1 and F be strictly a:s:
increasing. Then X i;n ! F 1 ðaÞ, where F 1 ðaÞ is the a-quantile of F. It then follows that for 0oaobo1
Corresponding author.
E-mail addresses: [email protected] (N. Balakrishnan), [email protected] (A. Stepanov). 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.08.001
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and i ¼ ½an þ 1; j ¼ ½bn þ 1; a:s:
Zi;j;n !
F 1 ðbÞ . F 1 ðaÞ
In the above expression, we suppose that F 1 ðaÞa0. Naturally, in the study of the asymptotic behaviour of Z i;j;n , interesting cases arise when i=n ! 1;
j=n ! 1; n ! 1,
(1.1)
i=n ! 0;
j=n ! 0; n ! 1.
(1.2)
Here, we focus on the case in (1.1) while the results for the case in (1.2) can be obtained in an analogous manner. a:s Observe that if the conditions in (1.1) hold, rF ¼ inffx : F ðxÞ ¼ 1go1 and rF a0, then Z i;j;n ! 1. This leads us to suppose for the main part of the paper that rF ¼ 1. However, we will also discuss later the special case when rF ¼ 0. While investigating the asymptotic behavior of Z i;j;n when the support is unbounded, we will see that the behaviour depends on the limiting properties of the distribution tail F¯ ðxÞ ¼ 1 F ðxÞ and can therefore be determined by the theory of regular variation. For the case when rF ¼ 0, we will find that the properties of slowly, regularly and rapidly varying functions at zero determine the asymptotic behaviour of the ratio Z i;j;n . This paper is organized as follows. In Section 2, some distributional results for order statistics as well as the notion of the theory of regular variation are presented. In Section 3, weak and strong limit theorems for Z nk;nkþl;n ð0olpk; n ! 1Þ are obtained. In Section 4, we study the asymptotic behaviour of Z i;j;n in the case when i=n ! 1; j=n ! 1; j i ! 1; n j ! 1: In Sections 3 and 4, results are developed under the assumption that rF ¼ 1. In Section 5, the results obtained in the preceding sections are developed for the case rF ¼ 0. The results developed in this paper are finally illustrated with some examples in Section 6. 2. Preliminaries 2.1. Distributional results for order statistics The distribution F i;n ðxÞ of X i;n is given by [see, for example, Arnold et al. (1992)] Z x n! F i;n ðxÞ ¼ F i1 ðuÞð1 F ðuÞÞni dF ðuÞ. ði 1Þ!ðn iÞ! 1
(2.1)
It readily follows from (2.1) that 1 X n¼kþ1
dF nk;n ðxÞ ¼ ðk þ 1Þ
dF ðxÞ . ½F¯ ðxÞ2
(2.2)
A conditional approach, based on the following lemma, is often useful for studying the properties of order statistics. Lemma 2.1. Under the condition X i;n ¼ x, the random vectors X ð1Þ ¼ ðX 1;n ; . . . ; X i1;n Þ and
X ð2Þ ¼ ðX iþ1;n ; . . . ; X n;n Þ ð1oionÞ
are conditionally independent; see, for example, David and Nagaraja (2003). Furthermore, the conditional distribution of X ð1Þ is the same as the unconditional distribution of the order statistics Y 1;i1 ; . . . ; Y i1;i1 arising from independent and identically distributed random variables Y 1 ; . . . ; Y i1 with the right-truncated distribution ð2Þ F ð1Þ is the same as the unconditional distribution of x ðuÞ ¼ F ðuÞ=F ðxÞ ðuoxÞ. The conditional distribution of X the order statistics W 1;ni ; . . . ; W ni;ni arising from independent and identically distributed random variables W 1 ; . . . ; W ni with the left-truncated distribution F ð2Þ x ðuÞ ¼ ðF ðuÞ F ðxÞÞ=ð1 F ðxÞÞ ðu4xÞ.
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2.2. Notion of the theory of regular variation In the case when rF ¼ inffx : F ðxÞ ¼ 1g ¼ 1, let us consider the limit lim
x!1
F¯ ðzxÞ ¼ bðzÞ 2 ½0; 1 F¯ ðxÞ
(2.3)
ðz41Þ.
It is known from the theory of regular variation that if the limit in (2.3) exists for any z41, then only the following three cases are possible: 1. if bðzÞ ¼ 0 for all z41, then F¯ is said to be a rapidly varying function; 2. if bðzÞ ¼ zg for all z41, where g40, then F¯ is said to be a regularly varying function with index g; and 3. if bðzÞ ¼ 1 for all z41, then F¯ is said to be a slowly varying function. For details, one may refer to Bingham et al. (1987, p. 52). In the case when rF ¼ 0, one may consider the limit F¯ ðzxÞ ¼ bðzÞ 2 ½0; 1 ¯ ðxÞ x!0 F lim
ð0ozo1Þ.
The above classification of rapidly, regularly and slowly varying functions continues here for functions varying at zero, with a small difference for regularly varying function; here, we have bðzÞ ¼ zg ð0ozo1; g40Þ. 3. Asymptotic behaviour of Z nk;nkþl;n when 0olpk are fixed In this and the following section, we assume rF ¼ 1. Using the properties of rapidly, regularly and slowly varying functions, one can get the following limit results for the variable Znk;nkþl;n when 0olpk are fixed and n ! 1. Theorem 3.1. (1) If F¯ is a rapidly varying function and 0olpk, then p
Z nk;nkþl;n ! 1 ðn ! 1Þ.
(3.1)
(2) If F¯ is a regularly varying function with index g40 and 0olpk, then l1 X k PfZ nk;nkþl;n 4zg ! ð1 zg Þi zgðkiÞ ðz41Þ. i i¼0
(3.2)
(3) If F¯ is a slowly varying function and 0olpk, then p
Z nk;nkþl;n ! 1
(3.3)
ðn ! 1Þ.
Proof of Theorem 3.1. Let us first prove Part (2) of the theorem. For any positive x0 and z41, we can write PfZ nk;nkþl;n 4zg ¼ I 1 ðk; l; n; x0 ; zÞ þ I 2 ðk; l; n; x0 ; zÞ Z x0 X nkþl;n ¼ 4zX nk;n ¼ x dF nk;n ðxÞ P x 1 Z 1 þ PfX nkþl;n 4zx j X nk;n ¼ xg dF nk;n ðxÞ. x0
For any fixed x0 ; z and k I 1 ðk; l; n; x0 ; zÞpF nk;n ðx0 Þ o
n!ð1 F ðx0 ÞÞk ðn k 1Þ!k!
Z
x0 1
F nk1 ðuÞ dF ðuÞo
nk nk F ðx0 Þ ! 0 ðn ! 1Þ. k!
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Applying Lemma 2.1 now, we have i ki l1 Z 1 X k F¯ ðzxÞ F¯ ðzxÞ I 2 ðk; l; n; x0 ; zÞ ¼ 1 dF nk;n ðxÞ. i F¯ ðxÞ F¯ ðxÞ x0 i¼0 For any small enough e40, a positive x0 can be chosen such that the integrands in I 2 ðk; l; n; x0 ; zÞ are bounded from above by ð1 zg þ eÞi ðzg þ eÞki . In the same way, one can show that these integrands are bounded from below by ð1 zg eÞi ðzg eÞki . These imply the statement in (3.2). Now, let us prove Parts (1) and (3) of the theorem. Applying Lemma 2.1 again, we have k Z ¯ F ðzxÞ PfZ nk;nkþ1;n 4zg ¼ dF nk;n ðxÞ. (3.4) F¯ ðxÞ R Using the same argument as in the proof of Part (2), we can obtain from (3.4) that for any fixed kX1 for rapidly (or slowly) varying distribution tail PfZ nk;nkþ1;n 4zg ! 0 ðor 1Þ, i.e. p
Znk;nkþ1;n ! 1
(3.5)
ðor 1Þ.
We obtain (3.1) and (3.3) from (3.5) by making use of the representation Znk;nkþj;n ¼ Z nk;nkþ1;n Z nkþ1;nkþ2;n Z nkþj1;nkþj;n
(3.6)
and the inequality Znk;nkþ1;n pZ nk;nkþl;n respectively.
ðlX1Þ,
(3.7)
&
The above theorem can be used to make the following observations. When F¯ is rapidly varying, the distribution tail is ‘‘thin’’ and so the distance between ‘‘top’’ order statistics X nk;n and X nkþl;n is ‘‘small’’, and consequently their ratio tends in probability to one. When F¯ is slowly varying, the distribution tail is ‘‘thick’’, and so the distance between X nk;n and X nkþl;n is ‘‘large’’, and consequently their ratio tends in probability to infinity. Imposing additional conditions on F¯ , we can obtain the following strong limit results for Z nk;nkþl;n . Theorem 3.2. (1) Let for any z41 and some kX1, k Z ¯ dF ðxÞ F ðzxÞ o1. ¯ F ðxÞ ½F¯ ðxÞ2 R
(3.8)
Then for all i and l such that iXk; 1plpi k þ 1, a:s
Zni;niþl;n ! 1. (2) Let for any z41 and some kX1, k # Z " dF ðxÞ F¯ ðzxÞ 1 o1. F¯ ðxÞ ½F¯ ðxÞ2 R Then for all i and l such that 1plpipk, a:s
Zni;niþl;n ! 1. Proof of Theorem 3.2. Let us first prove Part (1) of the theorem. Using Eqs. (2.2) and (3.4), we have k Z ¯ 1 X dF ðxÞ F ðzxÞ PfZ nk;nkþ1;n 4zg ¼ . ¯ F ðxÞ ½ F¯ ðxÞ2 R n¼kþ1
(3.9)
ARTICLE IN PRESS N. Balakrishnan, A. Stepanov / Statistics & Probability Letters 78 (2008) 301–310
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From the first part of Borel–Cantelli Lemma, it then follows that, under (3.8), Z nk;nkþ1;n ! 1. From (3.6) and the inequality k i F¯ ðzxÞ F¯ ðzxÞ X F¯ ðxÞ F¯ ðxÞ
(3.10)
ðiXkÞ,
a:s:
we obtain that Z ni;niþj;n ! 1 for iXk and 0ojpi k þ 1. The statement of Part (1) of the theorem therefore holds true. Next, from Eqs. (2.2) and (3.4), we have k # Z " 1 X dF ðxÞ F¯ ðzxÞ PfZ nk;nkþ1;n ozg ¼ 1 o1. ¯ F ðxÞ ½ F¯ ðxÞ2 R n¼kþ1 Then, Part (2) of the theorem now follows from Eqs. (3.7), (3.10) and Borel–Cantelli Lemma.
&
Remark 3.1. Observe that if condition (3.8) (or (3.9)) holds true, then F¯ is a rapidly (or slowly) varying function. 4. Asymptotic behaviour of Z i;j;n when j i; n j ! 1 As was already shown in Theorem 3.1, when F¯ is a slowly varying function, i.e., the distribution tail is p ‘‘thick’’, the distance between ‘‘top’’ order statistics X nk;n and X nkþl;n increases such that Znk;nkþl;n ! 1 for fixed 0olpk and n ! 1. This property of the ratio of order statistics holds as the order statistics are more distant. Thus, if F¯ is slowly varying and i=n ! 1, j=n ! 1, j i ! 1, then p
Z i;j;n ! 1.
(4.1)
The convergence in (4.1) holds when n j is either fixed or tends to infinity. Developing these ideas further, we can establish more interesting result. It is natural to suppose that if the distance between X i;n and X j;n increases in a proper speed, i.e., j i increases rapidly in comparison to n (or to n j), then Z i;j;n can also tend to infinity in probability. This is established in the following theorem. Theorem 4.1. Let F¯ be a slowly or a regularly varying function. Let i and j depend on n and i, j, n ! 1 such that i=n ! 1, j=n ! 1; n i ! 1; n j ! 1 and n j=n i ! 0. Then, p
Z i;j;n ! 1. Proof of Theorem 4.1. For proving this theorem, we need the following lemma taken from Dembinska et al. (2007). Lemma 4.1. Let i depend on n such way that n; i; n i ! 1 and i=n ! 0. Then, for any fixed 0oeo1, n! eni ! 0. ðn iÞ!ði 1Þ! For establishing the theorem, it is evident that it is sufficient to check the result for regularly varying F¯ . For any fixed positive x0 , PfZ i;j;n ozg ¼ I 1 ði; j; n; x0 ; zÞ þ I 2 ði; j; n; x0 ; zÞ Z 1 Z x0 X j;n ozX i;n ¼ x dF i;n ðxÞ þ P PfX j;n ozx j X i;n ¼ xgdF i;n ðxÞ. ¼ x x0 1 p
Under the conditions of Theorem 4.1, we have X i;n ! 1. Then, I 1 ði; j; n; x0 ÞpF i;n ðx0 Þ ! 0.
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Using Lemma 2.1 once again, we have ji1 nj Z zx Z 1 ðn iÞ! dF ðyÞ F¯ ðyÞ F¯ ðyÞ dF i;n ðxÞ 1 I 2 ði; j; n; x0 Þ ¼ ¯ ¯ ðn jÞ!ðj i 1Þ! x0 F¯ ðxÞ F ðxÞ F ðxÞ x ji Z 1 ðn iÞ! F¯ ðzxÞ 1 dF i;n ðxÞ. p ðn jÞ!ðj iÞ! x0 F¯ ðxÞ For any fixed z41 and any small enough e40, choose an x0 such that 1
F¯ ðzxÞ o1 zg þ e ¼ do1 F¯ ðxÞ
ðx4x0 Þ.
An upper bound for I 2 is then obtained as I 2 ði; j; n; x0 Þo
ðn iÞ! dji . ðn jÞ!ðj iÞ!
(4.2)
Let us now set ne ¼ n i and e i ¼ n j in (4.2). Observe that under the conditions of Theoremp 4.1, ne; e i; ne e e i ! 1 and i=e n ! 0. Applying now Lemma 4.1, we get I 2 ði; j; n; x0 ; zÞ ! 0, which means Z i;j;n ! 1: Hence, the theorem. & It is known from Theorem 3.1 that if F¯ is a rapidly varying function, then the ratio of ‘‘top’’ order statistics X nkþl;n and X nk;n tends in probability to one. We establish some more results in the following theorem for rapidly and regularly varying F¯ . Theorem 4.2. Let F¯ be a rapidly or regularly varying function. Let i and j depend on n and i; j; n ! 1 such that i=n ! 1; j=n ! 1; n j ! 1; j i ! 1 and n j=n i ! 1 ðor j i=n i ! 0). Then, p
Zi;j;n ! 1. Proof of Theorem 4.2. Obviously, it is sufficient to prove the theorem for regularly varying F¯ . As in the proof of Theorem 4.1, for some fixed positive x0 , we have Z PfZ i;j;n 4z j X i;n ¼ xg dF i;n ðxÞ PfZ i;j;n 4zg ¼ R Z 1 ¼ oð1Þ þ PfX j;n 4zx j X i;n ¼ xgdF i;n ðxÞ. x0
Then, ðn iÞ! PfZ i;j;n 4zg ¼ oð1Þ þ ðn jÞ!ðj i 1Þ!
Z
dF i;n ðxÞ ¯ R F ðxÞ
Z
1
zx
ji1 nj F¯ ðuÞ F¯ ðuÞ 1 dF ðuÞ. F¯ ðxÞ F¯ ðxÞ
We have ðn iÞ! PfZ i;j;n 4zgpoð1Þ þ ðn j þ 1Þ!ðj i 1Þ!
Z
1 x0
njþ1 F¯ ðzxÞ dF i;n ðxÞ. F¯ ðxÞ
For any fixed z41 and any small enough e40, choose a positive x0 such that F¯ ðzxÞ ozg þ e ¼ do1 F¯ ðxÞ
ðx4x0 Þ.
Then, PfZ i;j;n 4zgpoð1Þ þ
ðn iÞ! dnjþ1 . ðn j þ 1Þ!ðj i 1Þ!
(4.3)
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In the last inequality, let us set n i ¼ ne and j i 1 ¼ e i. Under the conditions of Theorem 4.2, e i; ne; ne e i! 1 and e i=e n ! 0. Applying now Lemma 4.1, we get PfZ i;j;n 4zg ! 0 for any fixed z41, which completes the proof of the theorem. & The following remark readily follows from the proof of Theorem 4.2. Remark 4.1. Let F¯ be a rapidly or regularly varying function. Let j iX1 be fixed and i; j; n ! 1 such that i=n ! 1; n j ! 1. Then, p
Z i;j;n ! 1. It is natural to suppose that under suitable stricter conditions, Theorems 4.1 and 4.2 can be transformed into strong limit results. Theorem 4.3. Let F¯ be a slowly or a regularly varying function. Let i and j depend on n and i; j; n ! 1 such that i=n ! 1; j=n ! 1; n i ! 1; n j ! 1 and n j=n i ! 0. If for some large enough fixed l 1 X
eðniÞ o1,
n¼l
then a:s:
Z i;j;n ! 1. Proof of Theorem 4.3. The following result, obtained by Dembinska et al. (2007) while proving Lemma 4.1, is required to prove this theorem. Lemma 4.2. Under the conditions of Lemma 4.1, we have n! 1 1 eni p pffiffiffiffiffiffi eðniÞðoð1Þþð log eÞÞ pffiffiffiffiffiffi enðoð1Þþð log eÞÞ . ðn iÞ!ði 1Þ! 2p 2p P P n ni Observe that because 1 converges, 1 also converges under the conditions n¼l e n¼l ðn!=ðn iÞ!ði 1Þ!Þe of Lemma 4.1. From the proof of Theorem 4.1, it follows that for some fixed positive x0 and any large enough fixed l, 1 X
PfZ i;j;n ozg ¼
n¼l
1 X
I 1 ði; j; k; x0 ; zÞ þ
n¼l
1 X
I 2 ði; j; k; x0 ; zÞ.
n¼l
As was shown earlier in the proof of Theorem 4.1, 1 X
I 1 ði; j; k; x0 ; zÞp
n¼l
1 X
F i;n ðx0 Þ.
n¼l
From (2.1), it is easily seen that 1 X n¼l
F i;n ðx0 Þp
1 X n¼l
n! F i ðx0 Þ. i!ðn iÞ!
In the series on the right-hand side of the above inequality, let us set ne ¼ n and e i ¼ n i. Observe that e e e e e i; n ; n i ! 1 and i=e n ! 0 which are the conditions of Lemma 4.1. So, from Lemma 4.2, it follows that P1 I ði; j; k; x ; zÞo1. 0 n¼l 1 It was already shown in the proof of Theorem 4.1 that for properly chosen x0 1 X n¼l
I 2 ði; j; n; x0 ; zÞp
1 X n¼l
ðn iÞ! dji . ðn jÞ!ðj iÞ!
(4.4)
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Let us now set ne ¼ n i P and e i ¼ n j. Observe that ne; e i; ne e i ! 1 and e i=e n ! 0. Recall that i ¼ iðnÞ. Let us denote e l ¼ l iðlÞX1. If 1 eeen o1, then the series in (4.4) converges and so the statement of the theorem en¼l holds true. & Following the same method of proof of Theorem 4.3, we can establish the following result. Theorem 4.4. Let F¯ be a rapidly or regularly varying function. Let i; j; n ! 1 such that i=n ! 1; n j ! 1; j i ! 1 and n j=n i ! 1 ðor j i=n i ! 0). If for some large enough fixed l 1 X eðnjÞ o1, n¼l
then a:s:
Zi;j;n ! 1. 5. The case rF ¼ 0 We have seen that the asymptotic properties of Z i;j;n in the case when rF ¼ 1 depend on the behaviour of F at infinity. One of the traditional ways to analyse this behaviour is by means of the theory of regular variation. Thus, in Sections 3 and 4, we established limit results for Z i;j;n in terms of rapidly, regularly and slowly varying functions at infinity. It is reasonable to expect that the asymptotic behaviour of Z i;j;n in the case when rF ¼ 0 is determined by the behaviour of F in the left vicinity of zero and can therefore be explained in terms of regularly, slowly and rapidly varying functions at zero. For this reason, let us consider the limit lim
x!0
F¯ ðzxÞ ¼ bðzÞ 2 ½0; 1 F¯ ðxÞ
ð0ozo1Þ.
All the results obtained earlier in Sections 3 and 4 can be easily transformed for this case. We present here only some of those results to show the slight difference. We present here the analogues of Theorems 3.1 and 3.2, for example, while other results of Sections 3 and 4 can be transformed in a similar manner. We do not present the proofs since they can be constructed from the corresponding proofs with slight modifications. Theorem 5.1. (1) If F¯ is a rapidly varying function at zero and 0olpk, then p
Znk;nkþl;n ! 1 ðn ! 1Þ. (2) If F¯ is a regularly varying function at zero with index g40 and 0olpk, then l1 X k ð1 zg Þi zgðkiÞ ð0ozo1Þ. PfZ nk;nkþl;n ozg ! i i¼0 (3) If F¯ is a slowly varying function at zero and 0olpk, then p
Znk;nkþl;n ! 0
ðn ! 1Þ.
Theorem 5.2. (1) Let for any 0ozo1 and some kX1, k Z 0 ¯ dF ðxÞ F ðzxÞ o1. ¯ F ðxÞ ½ F¯ ðxÞ2 1 Then, for all i and j such that iXk, 1pjpi k þ 1, a:s:
Zni;niþj;n ! 1. (2) Let for any 0ozo1 and some kX1, k # Z 0 " dF ðxÞ F¯ ðzxÞ 1 o1. ¯ F ðxÞ ½ F¯ ðxÞ2 1
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Then, for all i and j such that 1pjpipk, a:s:
Z ni;niþj;n ! 0. 6. Illustrative examples The results developed in the preceding sections are all illustrated with some examples in this section. pffiffiffiffiffiffi R x 2 x2 =2 ¯ 1. Consider the Normal Law FðxÞ ¼ ð1= 2pÞ 1 eu =2 du. Using the well-known expression FðxÞe = pffiffiffiffiffiffi ¯ is a rapidly varying function. Then, 2pxp[see, for example, Johnson et al. (1994)], we can show that F Z i;j;n ! 1 when either n i; n j ðn jon iÞ are finite, or when n i ! 1; n j ! 1 ðn jon iÞ and n j=n i ! 1. For large enough x0 , the integral tail Z
1
x0
k ¯ dFðxÞ FðzxÞ , 2 ¯ ¯ FðxÞ ½FðxÞ
behaves like pffiffiffiffiffiffi Z 1 2p 2 2 x2 eðx =2Þðkz k1Þ dx. z k x0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For kX1 and 1ozp 1 þ 1=k the last integral diverges and condition (3.8) in Theorem 3.2 does not hold. However, Theorem 4.4 can be applied here for Zi;j;n . a 2. Consider the distribution F ðxÞ ¼ 1 eðlog xÞ ðx41; a40Þ. For large x, we have F¯ ðzxÞ a log zðlog xÞa1 . e F¯ ðxÞ (a) For a41, the distribution tail is a rapidly varying function. Condition in (3.8) does not hold here either. All the statements made above in Example 1 for Z i;j;n are valid in this case as well. (b) For a ¼ 1, the tail is regularly varying with index 1. Then, PfZ nk;nkþl;n 4zg !
l1 X k i¼0
i
ð1 zÞi zðkiÞ
ð0olpk; n ! 1Þ.
Theorems 4.1–4.4 can also be applied here for Zi;j;n . p (c) When 0oao1, the distribution tail F¯ is slowly varying. Then, Z nk;nkþl;n ! 1. For large x0 and any kX1 the integral tail k # Z 1" dx F¯ ðzxÞ 1 , ¯ ¯ F ðxÞ ½F ðxÞ2 x0 behaves like Z
1
ak log z x0
ðlog xÞ2ða1Þ elog x
ax
dx.
That way, condition (3.9) does not hold. However, Theorems 4.1 and 4.3 can be applied here for Z i;j;n . x 3. Consider the extreme value distribution F ðxÞ ¼ 1 ee ðx 2 RÞ. We have F¯ ðzxÞ zx x ¼ ee þe ! 0 F¯ ðxÞ
ðx ! 1; z41Þ.
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Then F¯ is a rapidly varying function. Furthermore, Z ¯ F ðzxÞ dF ðxÞo1 ¯ ðxÞ3 R ½F a:s:
and it follows from (3.8) that Z nk;nkþl;n ! 1 for all kX1 and 0olpk as n ! 1. x!0 a 4. Consider the Weibull distribution F ðxÞ ¼ eðxÞ ðxo0; a40Þ. We have F¯ ðzxÞ=F¯ ðxÞ ! za ð0ozo1Þ: The distribution tail is regularly varying at zero with g ¼ a. Then, l1 X k ð1 za Þi zaðkiÞ ð0olpkÞ. PfZ nk;nkþl;n ozg ! i i¼0 P ni Let i; j; n ! 1 and n i; n j; j i ! 1. If n j=n i ! 0 ðand 1 o1Þ, then Zni;niþl;n n¼l e P1 nj p p a:s: a:s: ! 0ðor ! 0Þ. If n j=n i ! 1ðand n¼l e o1Þ, then Z ni;niþl;n ! 1 ðor ! 1Þ. Acknowledgement We are grateful to the referee for friendly and interesting remarks. One of such remarks has corrected an error in the paper. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Balakrishnan, N., Pakes, A., Stepanov, A., 2005. On the number and sum of near record value observations. Adv. Appl. Probab. 37, 1–16. Bingham, N.H., Goldie, C., Teugels, J.F., 1987. Regular Variation. Cambridge University Press, Cambridge, England. David, H.A., Nagaraja, H.N., 2003. Order Statistics, third ed. Wiley, Hoboken, NJ. Dembinska, A., Stepanov, A., 2006. Limit theorems for the ratio of weak records. Statist. Probab. Lett. 76, 1454–1464. Dembinska, A., Stepanov, A., Wesolowski, J., 2007. How many observations fall in a neighbourhood of an order statistic. Commun. Statis. Theory Methods 36, 851–867. Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous Univariate Distributions, vol. 1, second ed. Wiley, New York. Nevzorov, V.B., 2001. Records: Mathematical Theory, Translation of Mathematical Monographs, vol. 194. American Mathematical Society, Providence, RI. Shorrock, R.W., 1972. On record values and record times. J. Appl. Probab. 9, 316–326.
Statistics & Probability Letters 78 (2008) 301–310 www.elsevier.com/locate/stapro
Asymptotic properties of the ratio of order statistics N. Balakrishnana, A. Stepanovb, a
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West,Hamilton, Ont., Canada L8S4K1 b Department of Mathematics, Kaliningrad State Technical University, Sovietsky Prospect 1, Kaliningrad 236000, Russia Received 2 July 2007; received in revised form 2 August 2007; accepted 6 August 2007 Available online 17 August 2007
Abstract Let X 1;n pX 2;n p pX n;n be the order statistics obtained from a sample from a continuous distribution and Zi;j;n ¼ X j;n =X i;n be the ratio of order statistics. In this paper, asymptotic properties of Z i;j;n are discussed. Specifically, we make use of the theory of regular variation to derive weak and strong limit results for Zi;j;n . r 2007 Elsevier B.V. All rights reserved. MSC: 60G70; 62G30 Keywords: Order statistics; Ratio of order statistics; Limit laws; Theory of regular variation
1. Introduction The properties of order statistics have been studied quite extensively in the literature; one may refer to the books by Arnold et al. (1992), Nevzorov (2001), David and Nagaraja (2003), and the references therein. Yet, the asymptotic behaviour of the ratio of order statistics has not been thoroughly investigated. The method based on the theory of regular variation has been used for studying asymptotic properties of the ratio of record values and of weak record values recently by Balakrishnan et al. (2005) and Dembinska and Stepanov (2006), respectively; see also Shorrock (1972) for some early results in this direction. It appears that this method can be extended further and used for examining asymptotic properties of the ratio of order statistics. In this paper, we assume that X 1 ; X 2 ; . . . ; X n are independent and identically distributed random variables with a continuous distribution function F ðxÞ. Let X 1;n pX 2;n p pX n;n be the order statistics based on this sample, and Z i;j;n ¼ X j;n =X i;n ð1piojpnÞ be the ratio of order statistics. For the asymptotic behaviour of the ratio of order statistics, to begin with, we settle on the following aspects. Let i ¼ ½an þ 1, where ½ means the integral part of the number, 0oao1; n ! 1 and F be strictly a:s:
increasing. Then X i;n ! F 1 ðaÞ, where F 1 ðaÞ is the a-quantile of F. It then follows that for 0oaobo1
Corresponding author.
E-mail addresses: [email protected] (N. Balakrishnan), [email protected] (A. Stepanov). 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.08.001
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and i ¼ ½an þ 1; j ¼ ½bn þ 1; a:s:
Zi;j;n !
F 1 ðbÞ . F 1 ðaÞ
In the above expression, we suppose that F 1 ðaÞa0. Naturally, in the study of the asymptotic behaviour of Z i;j;n , interesting cases arise when i=n ! 1;
j=n ! 1; n ! 1,
(1.1)
i=n ! 0;
j=n ! 0; n ! 1.
(1.2)
Here, we focus on the case in (1.1) while the results for the case in (1.2) can be obtained in an analogous manner. a:s Observe that if the conditions in (1.1) hold, rF ¼ inffx : F ðxÞ ¼ 1go1 and rF a0, then Z i;j;n ! 1. This leads us to suppose for the main part of the paper that rF ¼ 1. However, we will also discuss later the special case when rF ¼ 0. While investigating the asymptotic behavior of Z i;j;n when the support is unbounded, we will see that the behaviour depends on the limiting properties of the distribution tail F¯ ðxÞ ¼ 1 F ðxÞ and can therefore be determined by the theory of regular variation. For the case when rF ¼ 0, we will find that the properties of slowly, regularly and rapidly varying functions at zero determine the asymptotic behaviour of the ratio Z i;j;n . This paper is organized as follows. In Section 2, some distributional results for order statistics as well as the notion of the theory of regular variation are presented. In Section 3, weak and strong limit theorems for Z nk;nkþl;n ð0olpk; n ! 1Þ are obtained. In Section 4, we study the asymptotic behaviour of Z i;j;n in the case when i=n ! 1; j=n ! 1; j i ! 1; n j ! 1: In Sections 3 and 4, results are developed under the assumption that rF ¼ 1. In Section 5, the results obtained in the preceding sections are developed for the case rF ¼ 0. The results developed in this paper are finally illustrated with some examples in Section 6. 2. Preliminaries 2.1. Distributional results for order statistics The distribution F i;n ðxÞ of X i;n is given by [see, for example, Arnold et al. (1992)] Z x n! F i;n ðxÞ ¼ F i1 ðuÞð1 F ðuÞÞni dF ðuÞ. ði 1Þ!ðn iÞ! 1
(2.1)
It readily follows from (2.1) that 1 X n¼kþ1
dF nk;n ðxÞ ¼ ðk þ 1Þ
dF ðxÞ . ½F¯ ðxÞ2
(2.2)
A conditional approach, based on the following lemma, is often useful for studying the properties of order statistics. Lemma 2.1. Under the condition X i;n ¼ x, the random vectors X ð1Þ ¼ ðX 1;n ; . . . ; X i1;n Þ and
X ð2Þ ¼ ðX iþ1;n ; . . . ; X n;n Þ ð1oionÞ
are conditionally independent; see, for example, David and Nagaraja (2003). Furthermore, the conditional distribution of X ð1Þ is the same as the unconditional distribution of the order statistics Y 1;i1 ; . . . ; Y i1;i1 arising from independent and identically distributed random variables Y 1 ; . . . ; Y i1 with the right-truncated distribution ð2Þ F ð1Þ is the same as the unconditional distribution of x ðuÞ ¼ F ðuÞ=F ðxÞ ðuoxÞ. The conditional distribution of X the order statistics W 1;ni ; . . . ; W ni;ni arising from independent and identically distributed random variables W 1 ; . . . ; W ni with the left-truncated distribution F ð2Þ x ðuÞ ¼ ðF ðuÞ F ðxÞÞ=ð1 F ðxÞÞ ðu4xÞ.
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2.2. Notion of the theory of regular variation In the case when rF ¼ inffx : F ðxÞ ¼ 1g ¼ 1, let us consider the limit lim
x!1
F¯ ðzxÞ ¼ bðzÞ 2 ½0; 1 F¯ ðxÞ
(2.3)
ðz41Þ.
It is known from the theory of regular variation that if the limit in (2.3) exists for any z41, then only the following three cases are possible: 1. if bðzÞ ¼ 0 for all z41, then F¯ is said to be a rapidly varying function; 2. if bðzÞ ¼ zg for all z41, where g40, then F¯ is said to be a regularly varying function with index g; and 3. if bðzÞ ¼ 1 for all z41, then F¯ is said to be a slowly varying function. For details, one may refer to Bingham et al. (1987, p. 52). In the case when rF ¼ 0, one may consider the limit F¯ ðzxÞ ¼ bðzÞ 2 ½0; 1 ¯ ðxÞ x!0 F lim
ð0ozo1Þ.
The above classification of rapidly, regularly and slowly varying functions continues here for functions varying at zero, with a small difference for regularly varying function; here, we have bðzÞ ¼ zg ð0ozo1; g40Þ. 3. Asymptotic behaviour of Z nk;nkþl;n when 0olpk are fixed In this and the following section, we assume rF ¼ 1. Using the properties of rapidly, regularly and slowly varying functions, one can get the following limit results for the variable Znk;nkþl;n when 0olpk are fixed and n ! 1. Theorem 3.1. (1) If F¯ is a rapidly varying function and 0olpk, then p
Z nk;nkþl;n ! 1 ðn ! 1Þ.
(3.1)
(2) If F¯ is a regularly varying function with index g40 and 0olpk, then l1 X k PfZ nk;nkþl;n 4zg ! ð1 zg Þi zgðkiÞ ðz41Þ. i i¼0
(3.2)
(3) If F¯ is a slowly varying function and 0olpk, then p
Z nk;nkþl;n ! 1
(3.3)
ðn ! 1Þ.
Proof of Theorem 3.1. Let us first prove Part (2) of the theorem. For any positive x0 and z41, we can write PfZ nk;nkþl;n 4zg ¼ I 1 ðk; l; n; x0 ; zÞ þ I 2 ðk; l; n; x0 ; zÞ Z x0 X nkþl;n ¼ 4zX nk;n ¼ x dF nk;n ðxÞ P x 1 Z 1 þ PfX nkþl;n 4zx j X nk;n ¼ xg dF nk;n ðxÞ. x0
For any fixed x0 ; z and k I 1 ðk; l; n; x0 ; zÞpF nk;n ðx0 Þ o
n!ð1 F ðx0 ÞÞk ðn k 1Þ!k!
Z
x0 1
F nk1 ðuÞ dF ðuÞo
nk nk F ðx0 Þ ! 0 ðn ! 1Þ. k!
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Applying Lemma 2.1 now, we have i ki l1 Z 1 X k F¯ ðzxÞ F¯ ðzxÞ I 2 ðk; l; n; x0 ; zÞ ¼ 1 dF nk;n ðxÞ. i F¯ ðxÞ F¯ ðxÞ x0 i¼0 For any small enough e40, a positive x0 can be chosen such that the integrands in I 2 ðk; l; n; x0 ; zÞ are bounded from above by ð1 zg þ eÞi ðzg þ eÞki . In the same way, one can show that these integrands are bounded from below by ð1 zg eÞi ðzg eÞki . These imply the statement in (3.2). Now, let us prove Parts (1) and (3) of the theorem. Applying Lemma 2.1 again, we have k Z ¯ F ðzxÞ PfZ nk;nkþ1;n 4zg ¼ dF nk;n ðxÞ. (3.4) F¯ ðxÞ R Using the same argument as in the proof of Part (2), we can obtain from (3.4) that for any fixed kX1 for rapidly (or slowly) varying distribution tail PfZ nk;nkþ1;n 4zg ! 0 ðor 1Þ, i.e. p
Znk;nkþ1;n ! 1
(3.5)
ðor 1Þ.
We obtain (3.1) and (3.3) from (3.5) by making use of the representation Znk;nkþj;n ¼ Z nk;nkþ1;n Z nkþ1;nkþ2;n Z nkþj1;nkþj;n
(3.6)
and the inequality Znk;nkþ1;n pZ nk;nkþl;n respectively.
ðlX1Þ,
(3.7)
&
The above theorem can be used to make the following observations. When F¯ is rapidly varying, the distribution tail is ‘‘thin’’ and so the distance between ‘‘top’’ order statistics X nk;n and X nkþl;n is ‘‘small’’, and consequently their ratio tends in probability to one. When F¯ is slowly varying, the distribution tail is ‘‘thick’’, and so the distance between X nk;n and X nkþl;n is ‘‘large’’, and consequently their ratio tends in probability to infinity. Imposing additional conditions on F¯ , we can obtain the following strong limit results for Z nk;nkþl;n . Theorem 3.2. (1) Let for any z41 and some kX1, k Z ¯ dF ðxÞ F ðzxÞ o1. ¯ F ðxÞ ½F¯ ðxÞ2 R
(3.8)
Then for all i and l such that iXk; 1plpi k þ 1, a:s
Zni;niþl;n ! 1. (2) Let for any z41 and some kX1, k # Z " dF ðxÞ F¯ ðzxÞ 1 o1. F¯ ðxÞ ½F¯ ðxÞ2 R Then for all i and l such that 1plpipk, a:s
Zni;niþl;n ! 1. Proof of Theorem 3.2. Let us first prove Part (1) of the theorem. Using Eqs. (2.2) and (3.4), we have k Z ¯ 1 X dF ðxÞ F ðzxÞ PfZ nk;nkþ1;n 4zg ¼ . ¯ F ðxÞ ½ F¯ ðxÞ2 R n¼kþ1
(3.9)
ARTICLE IN PRESS N. Balakrishnan, A. Stepanov / Statistics & Probability Letters 78 (2008) 301–310
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From the first part of Borel–Cantelli Lemma, it then follows that, under (3.8), Z nk;nkþ1;n ! 1. From (3.6) and the inequality k i F¯ ðzxÞ F¯ ðzxÞ X F¯ ðxÞ F¯ ðxÞ
(3.10)
ðiXkÞ,
a:s:
we obtain that Z ni;niþj;n ! 1 for iXk and 0ojpi k þ 1. The statement of Part (1) of the theorem therefore holds true. Next, from Eqs. (2.2) and (3.4), we have k # Z " 1 X dF ðxÞ F¯ ðzxÞ PfZ nk;nkþ1;n ozg ¼ 1 o1. ¯ F ðxÞ ½ F¯ ðxÞ2 R n¼kþ1 Then, Part (2) of the theorem now follows from Eqs. (3.7), (3.10) and Borel–Cantelli Lemma.
&
Remark 3.1. Observe that if condition (3.8) (or (3.9)) holds true, then F¯ is a rapidly (or slowly) varying function. 4. Asymptotic behaviour of Z i;j;n when j i; n j ! 1 As was already shown in Theorem 3.1, when F¯ is a slowly varying function, i.e., the distribution tail is p ‘‘thick’’, the distance between ‘‘top’’ order statistics X nk;n and X nkþl;n increases such that Znk;nkþl;n ! 1 for fixed 0olpk and n ! 1. This property of the ratio of order statistics holds as the order statistics are more distant. Thus, if F¯ is slowly varying and i=n ! 1, j=n ! 1, j i ! 1, then p
Z i;j;n ! 1.
(4.1)
The convergence in (4.1) holds when n j is either fixed or tends to infinity. Developing these ideas further, we can establish more interesting result. It is natural to suppose that if the distance between X i;n and X j;n increases in a proper speed, i.e., j i increases rapidly in comparison to n (or to n j), then Z i;j;n can also tend to infinity in probability. This is established in the following theorem. Theorem 4.1. Let F¯ be a slowly or a regularly varying function. Let i and j depend on n and i, j, n ! 1 such that i=n ! 1, j=n ! 1; n i ! 1; n j ! 1 and n j=n i ! 0. Then, p
Z i;j;n ! 1. Proof of Theorem 4.1. For proving this theorem, we need the following lemma taken from Dembinska et al. (2007). Lemma 4.1. Let i depend on n such way that n; i; n i ! 1 and i=n ! 0. Then, for any fixed 0oeo1, n! eni ! 0. ðn iÞ!ði 1Þ! For establishing the theorem, it is evident that it is sufficient to check the result for regularly varying F¯ . For any fixed positive x0 , PfZ i;j;n ozg ¼ I 1 ði; j; n; x0 ; zÞ þ I 2 ði; j; n; x0 ; zÞ Z 1 Z x0 X j;n ozX i;n ¼ x dF i;n ðxÞ þ P PfX j;n ozx j X i;n ¼ xgdF i;n ðxÞ. ¼ x x0 1 p
Under the conditions of Theorem 4.1, we have X i;n ! 1. Then, I 1 ði; j; n; x0 ÞpF i;n ðx0 Þ ! 0.
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Using Lemma 2.1 once again, we have ji1 nj Z zx Z 1 ðn iÞ! dF ðyÞ F¯ ðyÞ F¯ ðyÞ dF i;n ðxÞ 1 I 2 ði; j; n; x0 Þ ¼ ¯ ¯ ðn jÞ!ðj i 1Þ! x0 F¯ ðxÞ F ðxÞ F ðxÞ x ji Z 1 ðn iÞ! F¯ ðzxÞ 1 dF i;n ðxÞ. p ðn jÞ!ðj iÞ! x0 F¯ ðxÞ For any fixed z41 and any small enough e40, choose an x0 such that 1
F¯ ðzxÞ o1 zg þ e ¼ do1 F¯ ðxÞ
ðx4x0 Þ.
An upper bound for I 2 is then obtained as I 2 ði; j; n; x0 Þo
ðn iÞ! dji . ðn jÞ!ðj iÞ!
(4.2)
Let us now set ne ¼ n i and e i ¼ n j in (4.2). Observe that under the conditions of Theoremp 4.1, ne; e i; ne e e i ! 1 and i=e n ! 0. Applying now Lemma 4.1, we get I 2 ði; j; n; x0 ; zÞ ! 0, which means Z i;j;n ! 1: Hence, the theorem. & It is known from Theorem 3.1 that if F¯ is a rapidly varying function, then the ratio of ‘‘top’’ order statistics X nkþl;n and X nk;n tends in probability to one. We establish some more results in the following theorem for rapidly and regularly varying F¯ . Theorem 4.2. Let F¯ be a rapidly or regularly varying function. Let i and j depend on n and i; j; n ! 1 such that i=n ! 1; j=n ! 1; n j ! 1; j i ! 1 and n j=n i ! 1 ðor j i=n i ! 0). Then, p
Zi;j;n ! 1. Proof of Theorem 4.2. Obviously, it is sufficient to prove the theorem for regularly varying F¯ . As in the proof of Theorem 4.1, for some fixed positive x0 , we have Z PfZ i;j;n 4z j X i;n ¼ xg dF i;n ðxÞ PfZ i;j;n 4zg ¼ R Z 1 ¼ oð1Þ þ PfX j;n 4zx j X i;n ¼ xgdF i;n ðxÞ. x0
Then, ðn iÞ! PfZ i;j;n 4zg ¼ oð1Þ þ ðn jÞ!ðj i 1Þ!
Z
dF i;n ðxÞ ¯ R F ðxÞ
Z
1
zx
ji1 nj F¯ ðuÞ F¯ ðuÞ 1 dF ðuÞ. F¯ ðxÞ F¯ ðxÞ
We have ðn iÞ! PfZ i;j;n 4zgpoð1Þ þ ðn j þ 1Þ!ðj i 1Þ!
Z
1 x0
njþ1 F¯ ðzxÞ dF i;n ðxÞ. F¯ ðxÞ
For any fixed z41 and any small enough e40, choose a positive x0 such that F¯ ðzxÞ ozg þ e ¼ do1 F¯ ðxÞ
ðx4x0 Þ.
Then, PfZ i;j;n 4zgpoð1Þ þ
ðn iÞ! dnjþ1 . ðn j þ 1Þ!ðj i 1Þ!
(4.3)
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In the last inequality, let us set n i ¼ ne and j i 1 ¼ e i. Under the conditions of Theorem 4.2, e i; ne; ne e i! 1 and e i=e n ! 0. Applying now Lemma 4.1, we get PfZ i;j;n 4zg ! 0 for any fixed z41, which completes the proof of the theorem. & The following remark readily follows from the proof of Theorem 4.2. Remark 4.1. Let F¯ be a rapidly or regularly varying function. Let j iX1 be fixed and i; j; n ! 1 such that i=n ! 1; n j ! 1. Then, p
Z i;j;n ! 1. It is natural to suppose that under suitable stricter conditions, Theorems 4.1 and 4.2 can be transformed into strong limit results. Theorem 4.3. Let F¯ be a slowly or a regularly varying function. Let i and j depend on n and i; j; n ! 1 such that i=n ! 1; j=n ! 1; n i ! 1; n j ! 1 and n j=n i ! 0. If for some large enough fixed l 1 X
eðniÞ o1,
n¼l
then a:s:
Z i;j;n ! 1. Proof of Theorem 4.3. The following result, obtained by Dembinska et al. (2007) while proving Lemma 4.1, is required to prove this theorem. Lemma 4.2. Under the conditions of Lemma 4.1, we have n! 1 1 eni p pffiffiffiffiffiffi eðniÞðoð1Þþð log eÞÞ pffiffiffiffiffiffi enðoð1Þþð log eÞÞ . ðn iÞ!ði 1Þ! 2p 2p P P n ni Observe that because 1 converges, 1 also converges under the conditions n¼l e n¼l ðn!=ðn iÞ!ði 1Þ!Þe of Lemma 4.1. From the proof of Theorem 4.1, it follows that for some fixed positive x0 and any large enough fixed l, 1 X
PfZ i;j;n ozg ¼
n¼l
1 X
I 1 ði; j; k; x0 ; zÞ þ
n¼l
1 X
I 2 ði; j; k; x0 ; zÞ.
n¼l
As was shown earlier in the proof of Theorem 4.1, 1 X
I 1 ði; j; k; x0 ; zÞp
n¼l
1 X
F i;n ðx0 Þ.
n¼l
From (2.1), it is easily seen that 1 X n¼l
F i;n ðx0 Þp
1 X n¼l
n! F i ðx0 Þ. i!ðn iÞ!
In the series on the right-hand side of the above inequality, let us set ne ¼ n and e i ¼ n i. Observe that e e e e e i; n ; n i ! 1 and i=e n ! 0 which are the conditions of Lemma 4.1. So, from Lemma 4.2, it follows that P1 I ði; j; k; x ; zÞo1. 0 n¼l 1 It was already shown in the proof of Theorem 4.1 that for properly chosen x0 1 X n¼l
I 2 ði; j; n; x0 ; zÞp
1 X n¼l
ðn iÞ! dji . ðn jÞ!ðj iÞ!
(4.4)
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Let us now set ne ¼ n i P and e i ¼ n j. Observe that ne; e i; ne e i ! 1 and e i=e n ! 0. Recall that i ¼ iðnÞ. Let us denote e l ¼ l iðlÞX1. If 1 eeen o1, then the series in (4.4) converges and so the statement of the theorem en¼l holds true. & Following the same method of proof of Theorem 4.3, we can establish the following result. Theorem 4.4. Let F¯ be a rapidly or regularly varying function. Let i; j; n ! 1 such that i=n ! 1; n j ! 1; j i ! 1 and n j=n i ! 1 ðor j i=n i ! 0). If for some large enough fixed l 1 X eðnjÞ o1, n¼l
then a:s:
Zi;j;n ! 1. 5. The case rF ¼ 0 We have seen that the asymptotic properties of Z i;j;n in the case when rF ¼ 1 depend on the behaviour of F at infinity. One of the traditional ways to analyse this behaviour is by means of the theory of regular variation. Thus, in Sections 3 and 4, we established limit results for Z i;j;n in terms of rapidly, regularly and slowly varying functions at infinity. It is reasonable to expect that the asymptotic behaviour of Z i;j;n in the case when rF ¼ 0 is determined by the behaviour of F in the left vicinity of zero and can therefore be explained in terms of regularly, slowly and rapidly varying functions at zero. For this reason, let us consider the limit lim
x!0
F¯ ðzxÞ ¼ bðzÞ 2 ½0; 1 F¯ ðxÞ
ð0ozo1Þ.
All the results obtained earlier in Sections 3 and 4 can be easily transformed for this case. We present here only some of those results to show the slight difference. We present here the analogues of Theorems 3.1 and 3.2, for example, while other results of Sections 3 and 4 can be transformed in a similar manner. We do not present the proofs since they can be constructed from the corresponding proofs with slight modifications. Theorem 5.1. (1) If F¯ is a rapidly varying function at zero and 0olpk, then p
Znk;nkþl;n ! 1 ðn ! 1Þ. (2) If F¯ is a regularly varying function at zero with index g40 and 0olpk, then l1 X k ð1 zg Þi zgðkiÞ ð0ozo1Þ. PfZ nk;nkþl;n ozg ! i i¼0 (3) If F¯ is a slowly varying function at zero and 0olpk, then p
Znk;nkþl;n ! 0
ðn ! 1Þ.
Theorem 5.2. (1) Let for any 0ozo1 and some kX1, k Z 0 ¯ dF ðxÞ F ðzxÞ o1. ¯ F ðxÞ ½ F¯ ðxÞ2 1 Then, for all i and j such that iXk, 1pjpi k þ 1, a:s:
Zni;niþj;n ! 1. (2) Let for any 0ozo1 and some kX1, k # Z 0 " dF ðxÞ F¯ ðzxÞ 1 o1. ¯ F ðxÞ ½ F¯ ðxÞ2 1
ARTICLE IN PRESS N. Balakrishnan, A. Stepanov / Statistics & Probability Letters 78 (2008) 301–310
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Then, for all i and j such that 1pjpipk, a:s:
Z ni;niþj;n ! 0. 6. Illustrative examples The results developed in the preceding sections are all illustrated with some examples in this section. pffiffiffiffiffiffi R x 2 x2 =2 ¯ 1. Consider the Normal Law FðxÞ ¼ ð1= 2pÞ 1 eu =2 du. Using the well-known expression FðxÞe = pffiffiffiffiffiffi ¯ is a rapidly varying function. Then, 2pxp[see, for example, Johnson et al. (1994)], we can show that F Z i;j;n ! 1 when either n i; n j ðn jon iÞ are finite, or when n i ! 1; n j ! 1 ðn jon iÞ and n j=n i ! 1. For large enough x0 , the integral tail Z
1
x0
k ¯ dFðxÞ FðzxÞ , 2 ¯ ¯ FðxÞ ½FðxÞ
behaves like pffiffiffiffiffiffi Z 1 2p 2 2 x2 eðx =2Þðkz k1Þ dx. z k x0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For kX1 and 1ozp 1 þ 1=k the last integral diverges and condition (3.8) in Theorem 3.2 does not hold. However, Theorem 4.4 can be applied here for Zi;j;n . a 2. Consider the distribution F ðxÞ ¼ 1 eðlog xÞ ðx41; a40Þ. For large x, we have F¯ ðzxÞ a log zðlog xÞa1 . e F¯ ðxÞ (a) For a41, the distribution tail is a rapidly varying function. Condition in (3.8) does not hold here either. All the statements made above in Example 1 for Z i;j;n are valid in this case as well. (b) For a ¼ 1, the tail is regularly varying with index 1. Then, PfZ nk;nkþl;n 4zg !
l1 X k i¼0
i
ð1 zÞi zðkiÞ
ð0olpk; n ! 1Þ.
Theorems 4.1–4.4 can also be applied here for Zi;j;n . p (c) When 0oao1, the distribution tail F¯ is slowly varying. Then, Z nk;nkþl;n ! 1. For large x0 and any kX1 the integral tail k # Z 1" dx F¯ ðzxÞ 1 , ¯ ¯ F ðxÞ ½F ðxÞ2 x0 behaves like Z
1
ak log z x0
ðlog xÞ2ða1Þ elog x
ax
dx.
That way, condition (3.9) does not hold. However, Theorems 4.1 and 4.3 can be applied here for Z i;j;n . x 3. Consider the extreme value distribution F ðxÞ ¼ 1 ee ðx 2 RÞ. We have F¯ ðzxÞ zx x ¼ ee þe ! 0 F¯ ðxÞ
ðx ! 1; z41Þ.
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Then F¯ is a rapidly varying function. Furthermore, Z ¯ F ðzxÞ dF ðxÞo1 ¯ ðxÞ3 R ½F a:s:
and it follows from (3.8) that Z nk;nkþl;n ! 1 for all kX1 and 0olpk as n ! 1. x!0 a 4. Consider the Weibull distribution F ðxÞ ¼ eðxÞ ðxo0; a40Þ. We have F¯ ðzxÞ=F¯ ðxÞ ! za ð0ozo1Þ: The distribution tail is regularly varying at zero with g ¼ a. Then, l1 X k ð1 za Þi zaðkiÞ ð0olpkÞ. PfZ nk;nkþl;n ozg ! i i¼0 P ni Let i; j; n ! 1 and n i; n j; j i ! 1. If n j=n i ! 0 ðand 1 o1Þ, then Zni;niþl;n n¼l e P1 nj p p a:s: a:s: ! 0ðor ! 0Þ. If n j=n i ! 1ðand n¼l e o1Þ, then Z ni;niþl;n ! 1 ðor ! 1Þ. Acknowledgement We are grateful to the referee for friendly and interesting remarks. One of such remarks has corrected an error in the paper. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Balakrishnan, N., Pakes, A., Stepanov, A., 2005. On the number and sum of near record value observations. Adv. Appl. Probab. 37, 1–16. Bingham, N.H., Goldie, C., Teugels, J.F., 1987. Regular Variation. Cambridge University Press, Cambridge, England. David, H.A., Nagaraja, H.N., 2003. Order Statistics, third ed. Wiley, Hoboken, NJ. Dembinska, A., Stepanov, A., 2006. Limit theorems for the ratio of weak records. Statist. Probab. Lett. 76, 1454–1464. Dembinska, A., Stepanov, A., Wesolowski, J., 2007. How many observations fall in a neighbourhood of an order statistic. Commun. Statis. Theory Methods 36, 851–867. Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous Univariate Distributions, vol. 1, second ed. Wiley, New York. Nevzorov, V.B., 2001. Records: Mathematical Theory, Translation of Mathematical Monographs, vol. 194. American Mathematical Society, Providence, RI. Shorrock, R.W., 1972. On record values and record times. J. Appl. Probab. 9, 316–326.