Asymptotic normality of numbers of observations near order statistics from stationary processes

Statistics and Probability Letters 119 (2016) 259–263

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Asymptotic normality of numbers of observations near order statistics from stationary processes Krzysztof Jasiński Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland

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Article history: Received 2 July 2016 Received in revised form 19 August 2016 Accepted 23 August 2016 Available online 30 August 2016

abstract This paper is concerned with asymptotic normality of numbers of observations near order statistics. For some type of discontinuous marginal distributions, we extend knowing results to strictly stationary and ergodic observations satisfying an extra condition which guarantees some local independence. © 2016 Elsevier B.V. All rights reserved.

Keywords: Central limit theorems Extreme, intermediate, and central order statistics Near order statistic observations Stationary processes α -mixing condition

1. Introduction Let (Xn , n ≥ 1) be a sequence of random variables (rv’s) with a common cumulative distribution function (cdf) F . Denote by X1:n ≤ · · · ≤ Xn:n the order statistics based on the random sample (X1 , . . . , Xn ). If (kn , n ≥ 1) is a sequence of positive integers such that kn ≤ n for all n and kn /n → λ ∈ [0, 1] as n → ∞ then Xkn :n , n ≥ 1, are referred to as one of three natural cases: the case of central order statistics (corresponding to λ ∈ (0, 1)), that of extreme order statistics (when kn or n − kn is fixed) and that of intermediate order statistics (when λ ∈ {0, 1} and both kn and n − kn approach infinity). In this paper, we will study the asymptotic behavior of the following counting rv: Kkn :n (A) = #{j ∈ {1, . . . , n} : Xkn :n − Xj ∈ A}

(1.1)

as n → ∞, where A is a Borel subset of real numbers and 1 ≤ kn ≤ n. The rv in (1.1) provides information on how many observations fall into a random region determined by the set A and the order statistic Xkn :n . Exact and asymptotic properties of the rv Kkn :n (A) have been studied in the literature due to their applications to different practical problems. In particular, if kn = n and A = {0}, the rv in (1.1) counts elements in the sample of size n that are tied with the sample maximum. Moreover, if the cdf F is concentrated on nonnegative integers then Kn:n ({0}) can be interpreted as the numbers of winners in a game with n players whose scores are X1 , . . . , Xn . Under the assumption that (Xn , n ≥ 1) is a sequence of independent and identically distributed (i.i.d.) rv’s, numerous authors considered the existence of the limit of probability of no ties (limn→∞ Pr(Kn:n ({0})) = 1), and related problems. For more details see, for example, Eisenberg et al. (1993), Brands et al. (1994), Qi (1997), Bruss and Grübel (2003), Berred and Stepanov (2005), Eisenberg (2009) and Gouet et al. (2009).

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2016.08.014 0167-7152/© 2016 Elsevier B.V. All rights reserved.

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If F is continuous then Kkn :n ({0}) = 1 almost surely. So we do not obtain any information. Therefore, in that case Pakes and Steutel (1997) introduced the rv Kn:n ((0, a)) = #{j ∈ {1, . . . , n} : Xj ∈ (Xn:n − a, Xn:n )},

a > 0,

which counts the number of random points near the maximum. It is worth pointing out that this type of the rv can be used in actuarial mathematics to count insurance claims whose size is at prescribed distance from the existing largest claim size (see Li and Pakes, 2001 and Hashorva, 2003). Properties of the rv Kn:n ((0, a)) have been quite intensively studied. It was also generalized to the numbers of near r-extremes, Kn−r :n ((0, a)) and Kn−r :n ((−a, 0)), a > 0, 0 ≤ r ≤ n − 1. The extension was discussed in Pakes and Li (1998), Pakes (2000), Hu and Su (2003), Balakrishnan and Stepanov (2005), Dembińska and Balakrishnan (2008, 2010). Next, this concept was extended and the asymptotic properties of the numbers of near order statistic observations were obtained, Kkn :n ((0, a)) and Kkn :n ((−a, 0)), a > 0, kn /n → λ ∈ [0, 1] (see Dembińska et al., 2007; Pakes, 2009; Iliopoulos et al., 2012 for more details). Finally, these results were generalized to any Borel set A, Kkn :n (A) by Dembińska (2012a) and Dembińska and Iliopoulos (2012). Some generalizations to the case of not necessarily independent sequences of observations are given by Hashorva (2003), Hashorva and Hüsler (2004) and Balakrishnan et al. (2009). In the literature one problem, that was extensively investigated, is devoted to the question whether the rv Kkn :n (A) can be centered and normed to yield a normal limit law. The case of extreme order statistics and continuous F was discussed by Pakes and Steutel (1997), Hashorva and Hüsler (2004) and Pakes (2009) under the assumption that the left endpoint of the support of F is finite. For central order statistics and continuous F , these properties were studied by Pakes (2009), Iliopoulos et al. (2012) and Dembińska (2012a). Moreover, two situations: one where the cdf F is discontinuous and the order statistics are extreme, central, intermediate and the other where it is continuous and the order statistics are extreme, were considered by Dembińska (2012b). However, all these results were obtained under i.i.d. assumption of the sequence (Xn , n ≥ 1). The aim of this paper is to extend the results for discontinuous case given by Dembińska (2014b). We will replace the i.i.d. assumption with a weaker one that (Xn , n ≥ 1) is a strictly stationary and ergodic sequence. This extension is important because dependence is often observed in practical applications. We obtain our result while the cdf F is discontinuous, under some standard conditions which guarantee some local independence and hold for all three cases discussed in the literature: central, extreme and intermediate order statistics. Discontinuity here means that the corresponding λth quantile of the cdf F is not an accumulation point of its support. Throughout the paper we make use of the following notation. The rv’s Xn , n ≥ 1, exist in a probability space (Ω , F , P ). R and Z represent the sets of real numbers and integers, respectively. B (R) stands for the Borel σ -field of subsets of R. The support of the distribution F is denoted by supp(F ) and we set γ0 := inf supp(F ) = inf{x ∈ R : F (x) > 0}, γ1 := sup supp(F ) = sup{x ∈ R : F (x) < 1}. By γλ we denote the unique λth quantile of F where λ ∈ (0, 1). We write I (·) for the indicator function, that is I (x ∈ A) = 1 if x ∈ A and I (x ∈ A) = 0 otherwise; and we use IA as an d

a.s.

abbreviation for I (x ∈ A). Moreover, by −→ and −→ we denote convergence in distribution and almost sure convergence, respectively. 2. Main result Let (Xn , n ≥ 1) be a sequence of not necessarily independent but identically distributed rv’s with a common discontinuous cdf F . More precisely, we relax the standard assumption that Xn , n ≥ 1 are independent. We replace it by the weaker one that (Xn , n ≥ 1) is a strictly stationary and ergodic sequence. However, an extra condition is required to ensure that central, extreme, intermediate order statistics from such stationary and ergodic processes have similar asymptotic behavior as the respective ones from i.i.d. observations. This condition is called strong mixing (α -mixing) condition and guarantees a specific type of some local asymptotic independence. Before we formulate it, we introduce some notation. For any two σ -fields A and B ⊂ F , we define the measure of dependence

α (A, B ) :=

sup

A∈A, B∈B

| P(A ∩ B) − P(A) P(B)|.

If X = (Xk , k ∈ Z) is a two-sided random sequence, then the coefficients of the α -mixing condition are given by

  αX (n) := sup α σ (. . . , Xj−1 , Xj ), σ (Xj+n , Xj+n+1 , . . .) ,

n ≥ 1,

j∈Z

where σ (Xk , k ∈ J ), J ⊂ Z, is the σ -field generated by the rv’s Xk , k ∈ J. For a one-sided random sequence X = (Xn , n ≥ 1), αX (n) is by definition the same as for two-sided sequence of the form (. . . , 0, 0, 0, X1 , X2 , X3 , . . .). Under these notation, we say that X is strongly mixing (α -mixing), if αX (n) → 0 as n → ∞. For discussion of α -mixing condition see Bradley (2007). Before we present the main theorem that asserts the asymptotic normality of a centered and normed version of the rv Kkn :n (A) when the cdf F is discontinuous, we begin with the following result which describes the asymptotic behavior of the order statistics from stationary and ergodic sequences. It extends Lemma 1 of Dembińska (2012b), because the independence assumption is relaxed.

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Lemma 1. Let (Xn , n ≥ 1) be a strictly stationary and ergodic sequence of rv’s with an arbitrary cdf F and (kn , n ≥ 1) be a sequence of positive integers such that kn ≤ n for all n ≥ 1 and kn /n → λ ∈ [0, 1]. Then a.s.

(a) Xkn :n −→ γ0 as n → ∞ provided that λ = 0 and γ0 > −∞, a.s.

(b) Xkn :n −→ γ1 as n → ∞ provided that λ = 1 and γ1 < ∞, a.s.

(c ) Xkn :n −→ γ as n → ∞ if λ ∈ (0, 1). Proof. Part (c) is Lemma 2.1 of Dembińska (2014a) and is given for completeness. However, it gives the method of proof which remains valid for parts (a) and (b). We restrict ourselves to showing (b), because the proof of (a) goes along the same lines. First observe that since (Xn , n ≥ 1) is strictly stationary and ergodic, the sequence (I (Xn < γ1 − ε), n ≥ 1) is so as well. This is a consequence, for example, of Proposition 2.10 of Bradley (2007). Applying the strong ergodic theorem (see, for example, Grimmet and Stirzaker, 2004), under the assumption that γ1 < ∞, we get for any arbitrary but fixed ε > 0, n 

a.s.

I (Xi < γ1 − ε)/n −→ P(X1 < γ1 − ε) = F (γ1 − ε − ) < 1.

i =1

Hence

 P(Xkn :n ≥ γ1 − ε for all large n) = P

n 

 I (Xi < γ1 − ε) < kn for all large n

i =1

 n  i =1 = P 



I (Xi < γ1 − ε) n

<

kn n

for all large n  = 1,



which means that lim infn→∞ Xkn :n ≥ γ1 − ε a.s. Since ε > 0 was taken to be arbitrary, we obtain the inequality lim infn→∞ Xkn :n ≥ γ1 a.s. Moreover, we immediately get the relation lim supn→∞ Xkn :n ≤ γ1 a.s. as a consequence of the fact that, for any n, Xkn :n ≤ γ1 a.s. This gives (b) and completes the proof of the lemma.  Now, we are in a position to formulate the main result of this paper. Observe that it covers all the three cases studied in the literature: extreme, intermediate and central. In the first two cases, we have to assume that supp(F ) is bounded. In the third one we require the uniqueness of λth quantile. Below the discontinuity of the cdf F means that λth quantiles (including γ0 , γ1 ) are not accumulation points of supp(F ). Theorem 1. Suppose that X = (Xn , n ≥ 1) is a strictly stationary sequence of rv’s with discontinuous cdf F . Suppose that ∞ n=1 αX (n) < ∞ and (kn , n ≥ 1) is non-decreasing sequence of positive integers such that kn /n → λ ∈ [0, 1]. Moreover, assume that

• if λ = 0, then γ0 > −∞; • if λ = 1, then γ1 < ∞; • if λ ∈ (0, 1), then there exists a unique λth quantile γλ . Then, for any A ∈ B (R), as n → ∞,

√ n



Kkn :n (A) n

 d − P(X1 ∈ γλ − A) −→ N (0, σX2 ),

provided that γλ is not an accumulation point of supp(F ) and

σX2 = p(1 − p) + 2

∞  

P(X1 ∈ γλ − A, Xj ∈ γλ − A) − p2 ̸= 0,



(2.1)

j =2

where p = P(X1 ∈ γλ − A). Before we proceed to prove the above theorem, we give some helpful remarks. Firstly, it is easy to see that the condition ∞ n=1 αX (n) < ∞ implies αX (n) → 0, as n → ∞. Therefore, X = (Xn , n ≥ 1) is an α -mixing process. Lemma 2. Under the assumptions of Theorem 1, the random sequence Y = (Yn , n ≥ 1), where Yn = I (Xn ∈ γλ − A) − P(X1 ∈ γλ − A), n ≥ 1, satisfies the following conditions:

(a) E(Y1 ) = 0, (b) P(|Y1 | ≤ C ) = 1 for some constant C ,

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(c) Y is an α -mixing strictly stationary process, ∞  (d) αY (n) < ∞, where αY (n), n ≥ 1, are the α -mixing coefficients of the process Y. n=1

Proof. Parts (a) and (b) follow from the definition of the process Y. To proof (c), we first observe that Y can be obtained from X by some Borel function f : R → R, thus Yn = f (Xn ), n ≥ 1. Hence, Y is also a strictly stationary and α -mixing process with αY (n) ≤ αX (n) for each n ∈ N (see Remark 1.8(a) of Bradley, 2007). Now, by the majorant criterion of convergence of ∞ number series we get that n=1 αY (n) < ∞, which completes the proof of the lemma.  Now, using Lemma 2 and Theorem 10.3 of Bradley (2007) we obtain that σX ∈ [0, ∞). The next observation is a direct consequence of Theorem 2 and following its eleven lines on p. 1191 of Rio (1995). Remark 1. Lemma 2 implies that σX2 > 0. One can show even more, namely the process Y = (Yn , n ≥ 1) obeys the law of the iterated logarithm. Now we are ready to prove Theorem 1. Proof. Let

√ Un :=



n

Kkn :n (A) n

 − P(X1 ∈ γλ − A)

and Wn :=

n  Yi √ ,

n

i =1

where Y = (Yn , n ≥ 1) is a random sequence defined as Yn = I (Xn ∈ γλ − A) − P(X1 ∈ γλ − A), n ≥ 1. Lemma 2 and Remark 1 imply that all assumptions of the central limit theorem for bounded α -mixing sequences are d

satisfied (see Theorem 10.3 of Bradley, 2007). Therefore Wn −→ N (0, σX2 ), as n → ∞, where

σX2 = E Y12 + 2

∞ 

E (Y1 Yk ) .

k =2

By definition of the process Y, it is easy to check that σX2 is the same as in (2.1). Since X = (Xn , n ≥ 1) is strictly stationary and α -mixing, we obtain by Remark 2.6 of Bradley (2007) that is ergodic too. Moreover, since γλ is not an accumulation point of supp(F ), we conclude from Lemma 1 that P(Xkn :n = γλ for all large n) = 1. Hence, n  

I (Xi ∈ Xkn :n − A) − I (Xi ∈ γλ − A)

Un − Wn =

i=1

√

n

 = 0,

where the above equality holds almost surely for all large n. This yields a.s.

Un − Wn −→ 0 as n → ∞. By Slutsky’s lemma, d

Un = (Un − Wn ) + Wn −→ N (0, σX2 ), which completes the proof.



Theorem 1 can be applied ∞ to any sequence (Xn , n ≥ 1) which is strictly stationary and ergodic with a discontinuous cdf F for which the condition n=1 αX (n) < ∞ holds. Recall that the class of strictly stationary and ergodic sequences includes, besides sequences (Xn , n ≥ 1) of i.i.d. rv’s, for example linear processes, that is processes of the form (Xn , n ≥ 1) where Xn =

∞ 

ai εn−i

(2.2)

i=−∞

and (εk , k ∈ Z) is a two-sided sequence of i.i.d. rv’s and (ai , i ∈ Z) is a two-sided sequence of constants such that the sum in (2.2) is a.s. convergent. However, the fact that the widely used class of linear processes need not be α -mixing leads to a need to establish conditions under which a linear process is α -mixing. This problem was discussed by Chanda (1974), whose results were next corrected by Gorodetskii (1977). Alternative set of conditions was given by Withers (1981). The class of linear processes covers, among others, all stationary autoregressive-moving average (ARMA) processes. Their α -mixing properties were established by Athreya and Pantula (1986). For more details about some mixing properties of time series models, we refer also to Pham and Tran (1985).

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Remark 2. It is still an open question whether a similar result to Theorem 1 holds under the assumption that the cdf F is continuous. In other words, an unsolved problem is to find an extended version of Theorem 3.1 of Dembińska (2012a) for strictly stationary and ergodic sequences with an extra condition which guarantees a specific type of some local asymptotic independence. References Athreya, K.B., Pantula, S.G., 1986. A note on strong mixing of ARMA processes. Statist. Probab. Lett. 4, 187–190. Balakrishnan, N., Hashorva, E., Hüsler, J., 2009. Near-extremes and related point processes. Albanian J. Math. 3, 63–74. Balakrishnan, N., Stepanov, A., 2005. A note on the number of observations near an order statistics. J. Statist. Plann. Inference 134, 1–14. Berred, A., Stepanov, A., 2005. Ties for the second place. In: Ahsanullah, M., Raqab, M. (Eds.), Recent Developments in Ordered Random Variables. Nova Science Publisher, New York, pp. 171–185. Bradley, R.C., 2007. Introduction to Strong Mixing Conditions, Volume 1. Kendrick Press, Heber City (Utah). Brands, J.J.A.M., Steutel, F.W., Wilms, R.J.G., 1994. On the number of maxima of a discrete sample. Statist. Probab. Lett. 20, 209–218. Bruss, F.T., Grübel, R., 2003. On the multiplicity of the maximum in a discrete random sample. Ann. Appl. Probab. 13, 1252–1263. Chanda, K.C., 1974. Strong mixing properties of linear stochastic processes. J. Appl. Probab. 11, 401–408. Dembińska, A., 2012a. Asymptotic properties of numbers of observations in random regions determined by central order statistics. J. Statist. Plann. Inference 142, 516–528. Dembińska, A., 2012b. Limit theorems for proportions of observations falling into random regions determined by order statistics. Aust. N. Z. J. Stat. 54 (2), 199–210. Dembińska, A., 2014a. Asymptotic behavior of central order statistics from stationary sequences. Stochastic Process. Appl. 124, 348–372. Dembińska, A., 2014b. Asymptotic normality of numbers of observations in random regions determined by order statistics. Statistics 48, 508–523. Dembińska, A., Balakrishnan, N., 2008. The asymptotic distribution of numbers of observations near order statistics. J. Statist. Plann. Inference 138, 2552–2562. Dembińska, A., Balakrishnan, N., 2010. On the asymptotic independence of numbers of observations near order statistics. Statistics 44, 517–528. Dembińska, A., Iliopoulos, G., 2012. On the asymptotics of numbers of observations in random regions determined by order statistics. J. Multivariate Anal. 103, 151–160. Dembińska, A., Stepanov, A., Wesołowski, J., 2007. How many observations fall in a neighborhood of an order statistic? Comm. Statist. – Theory Methods 36, 851–867. Eisenberg, B., 2009. The numbers of players tied for the record. Statist. Probab. Lett. 79, 283–288. Eisenberg, B., Stengle, G., Strang, G., 1993. The asymptotic probability of a tie for first place. Ann. Appl. Probab. 3, 731–745. Gouet, R., López, F.J., Sanz, G., 2009. Limit laws for the cumulative number of ties for the maximum in a random sequence. J. Statist. Plann. Inference 139, 2988–3000. Grimmet, G.R., Stirzaker, D.R., 2004. Probability and Random Processes. Oxford University Press, New York. Gorodetskii, V.V., 1977. On the strong mixing property for linear sequences. Theory Probab. Appl. 22, 411–413. Hashorva, E., 2003. On the number of near-maximum insurance claim under dependence. Insurance Math. Econom. 32, 37–49. Hashorva, E., Hüsler, J., 2004. Estimation of tails and related quantities using the number of near-extremes. Comm. Statist. – Theory Methods 34, 337–349. Hu, Z., Su, C., 2003. Limit theorems for the number and sum of near-maxima for medium tails. Statist. Probab. Lett. 63, 229–237. Iliopoulos, G., Dembińska, A., Balakrishnan, A., 2012. Asymptotic properties of numbers of observations near sample quantiles. Statistics 46, 85–97. Li, Y., Pakes, A., 2001. On the number of near maximum insurance claims. Insurance Math. Econom. 28, 309–323. Pakes, A., 2000. The number and sum of near-maxima for thin-tailed populations. Adv. Appl. Probab. 32, 1100–1116. Pakes, A., 2009. Numbers of observations near order statistics. Aust. N. Z. J. Stat. 51, 375–395. Pakes, A., Li, Y., 1998. Limit laws for the number of near maxima via the Poisson approximation. Statist. Probab. Lett. 40, 395–401. Pakes, A., Steutel, F.W., 1997. On the number of records near the maximum. Austral. J. Statist. 39, 179–193. Pham, D.T., Tran, T.T., 1985. Some mixing properties of time series models. Stochastic Process. Appl. 19, 297–303. Rio, E., 1995. The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23, 1188–1203. Qi, Y., 1997. A note on the number of maxima in a discrete sample. Statist. Probab. Lett. 33, 373–377. Withers, C.S., 1981. Conditions for linear processes to be strong-mixing. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 57, 477–480.