Scholarly article on topic 'Precise Rates in Log Laws for NA Sequences'

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Academic research paper on topic "Precise Rates in Log Laws for NA Sequences"

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2007, Article ID 89107, 11 pages doi:10.1155/2007/89107

Research Article

Precise Rates in Log Laws for NA Sequences

Yuexu Zhao

Received 27 September 2006; Revised 23 December 2006; Accepted 30 January 2007

Let X1, X2,... be a strictly stationary sequence of negatively associated (NA) random variables with EX1 = 0, set Sn = X1 + ■■■ + Xn, suppose that a2 = EXf + 2]^ ^=2EX1Xn > 0 and EX2 < o, if -1 <a< 1; EX? (log |X1 |)a < o, if a > 1. We prove lim ei0e2a+2X ~=1((log n)a/ n)P(\Sn\> a(e+Kn)^2nlogn) = 2-(a+1)(a +1)-1E|N|2a+2, where Kn = O(1/logn) and N is the standard normal random variable.

Copyright © 2007 Yuexu Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A finite family of random variables, X1,X2,...,Xn, is said to be NA if, for every pair of disjoint subsets T1 and T2 of {1,2,..., n},

Cov (fKX,i e TO, f2(Xj, j e T2)) < 0, (1.1)

whenever f1 and f2 are coordinatewise increasing and the covariance exists. An infinite family is NA if every finite subfamily is NA. This definition was introduced by Alam and Saxena [1] and Joag-Dev and Proschan [2], and has found many applications in percolation theory, multivariate statistical analysis, and reliability theory (see, e.g., Barlow and Proschan [3]).

Let { X n : n > 1} be a sequence of NA random variables on some probability space (O, P) with mean zero and finite variance. As usual, set S0 = 0, Sn = X1 + ■■■ + Xn, n > 1, and write a2 = ES?. Under appropriate covariance conditions, many limit theorems have been obtained. For example, the central limit theorem was proved by Newman [4].

Theorem 1.1. Let {Xn : n > 1} be strictly stationary NA sequences with mean zero and

0 < a2 = EX2 + 2 £ EX1Xn < to, (1.2)

Sn 3 N(0,1), as n — to. (1.3)

Further results are three series theorems (see, e.g., Matula [5]), probability inequalities (cf. Roussas [6], Shao [7]), weak convergence (see, e.g., Zhang [8]), the complete convergence (cf. Liang and Su [9], Liang [10]), and the law of the iterated logarithm (see, e.g., Shao and Su [11], Zhang [12]), and so forth.

Note that in the above-mentioned limit theorems, the convergence rates of logarithm are little known, the purpose of the present paper is to investigate the precise asymptotics in the law of the logarithm for NA sequences. It is well known that NA sequences can contain independent random variables as special case, many authors have given lots of beautiful results for independent variables. Let us first recall parts of those results, it is very convenient to adopt the following notations: letX1,X2,... be independent and identically distributed (i.i.d.) nondegenerate random variables with EX1 = 0 and EXf = a2 < to, set Sn = X1 + ■■■ + Xn, logx = loge(x V e). Chow and Lai [13] studied the following results.

Theorem 1.2. Suppose that VarX1 = a2 and a > 1. Then the following are equivalent:

^ na-2p( | Sn | > e-^j2nlogn) < to, ye > a-Ja - 1;

to , _, _

^ na-2Pi max |Sk | > e-^2nlognl < to, ye > a-Ja - 1;

n=1 l<k<n

Xna-2^ |Sn| > e-\j2nlogn) < to, for some e > 0;

E|X |2a EX1 = 0, ^

Heyde [14] presented an interesting and beautiful result. Theorem 1.3. If EX1 = 0 and EX2 < to, then

lime2Y, P( | Sn | > en) = EX\. (1.5)

e*0 n=1

This is a precise estimate for the convergence rate of probability series as e \ 0, which has been generalized and extended in several directions. For a = 1 in Theorem 1.2, Gut and Spataru [15] obtained the results as follows.

Theorem 1.4. Suppose that EX\ = 0 and EX2 = a2 < o. Then, for 0 < S < 1,

^ f <!oiï>!p(|S„u = (1.6)

where N is a standard normal random variable.

Our starting point is Theorem 1.4, the present work will give the analogue of (1.6) for NA sequences. From now on, we adopt the following notations: let X1,X2,... be strictly stationary NA sequences with EX1 = 0 and EX? < n, a2 = EXf + 2 X0=2 EX1Xn > 0, and set Sn = X1 + ■ ■■ + Xn, Mn = max1<k<n |Sk |, write log for the natural logarithm, logx = loge(x V e), [z] denotes the largest integer which is not larger than z, C denotes positive constant, independent of e, it may take different values in each appearance. The paper is organized as follows: we first introduce our main results, after which the proofs of Theorems 2.1 and 2.4 are exposed in Sections 3 and 4, respectively. We now state the main results.

2. Main results

Theorem 2.1. Let Kn = 0(1/log n), EX2 < n, if -1 <a < 1; EX?(log X |)a < n, if a > 1. Then

lime2a+2 f (on)^P(|S„| > a(e + Kn2nlogn)

= 2-(a+1)(a + 1)-1E|N |2a+2,

where N is a standard normal random variable. Corollary 2.2. Under the conditions in Theorem 2.1,

lime2«+2 f p( 1 Sn 1 > ae JS^ = (2.2)

eio , n V1 n| V & ) 2(a+1)(a +1)

n=1 v '

Corollary 2.3. Suppose that EXj2 < o. Then

. ,-, EN 2

lim e2^ n-1P( | Sn | > a (e + Kn) ^2n log n) =-. (2.3)

e*0 n=1

Theorem 2.4. Let Kn = 0(1/log n), EX\ < o, if-1 <a < 1/2; EX2(log |X1 |)a < o, if a > 1/2. Then

lime2a+2 f (logn)a^Mn > a(e + kJJ2nlogn)

e*0 n=1 n

" (-1)n

= 2-a(a +1)-1E|N|2a+2 X

(2n +1)2a+2'

Without loss of generality, throughout the paper, we will suppose that a2 = 1. Let $(x) denote the standard normal distribution function, and put ¥(x) = 1 - $(x) + $(-x), x > 0.

3. Proof of Theorem 2.1

In order to prove this result easily, we separate the proof into two propositions, the first one can be formulated as follows.

Proposition 3.1. Suppose that N be a nondegenerate Gaussian random variable. Then

lime2a+2 X ^^p(\n\ >(e + Kn)Jlkg)

ei0 n=i n V V 7 (3.1)

= 2-(a+1)(a +1)-1£\N\2(a+1). Proof. Noting the definition of Kn, we first show that

lime2a+2 f (logn)"P(\N\ > eJ2logn) = 2-(a+1)(a +1)-1£\N\2a+2. (3.2)

el0 , n v v '

By integral formula and transformation, it is enough to show that for any a > -1,

lime2a+2 V P(\N\ > eJ2logn)

ei0 , n v v '

= lime2a+2 V f+1 (ogx)aP(\N\ > eJ2logx)dx

ei0 ^J n X V1 1 M 8 J (3.3)

= 2-a y2a+1P(\N\ > y)dy

= 2-(a+1)(a +1)-1£ \ N \ 2a+2.

An(e) = |p( \ N\ >eJ2logn) - p( \ N\ > (e + k^^2logn)|. (3.4)

The proof of (3.1) should be completed, if one could show that

lim Me) = 0, (3.5)

c i fl -1 *m

the proof of (3.5) is similar to that of Proposition 2.2 in Huang and Zhang [16].

Before giving the second proposition, the following lemma is necessary. □

Lemma 3.2 [17]. Suppose that {Xk : k > 1} be NA sequences with EXk = 0, E\Xk \p < oo, for p > 2. Then, for any t > p/2, x > 0,

P(\Sn\> x) < Jp(\Xk\> t) +2e^ 1+ t EX2 ^ • (3.6)

Proposition 3.3. Suppose that EX2 < oo, if-1 < a < 1; EX?(log \X1 \)a < oo, if a > 1. Then

lim e2a+2 J (^\p(\Sn\ > (e + Kn)yj 2n log n) - p(\N \ > (e + Kn)yj 2logn) \ = 0.

Proof. Set H(e) = [exp(M/e2)], where M > 4, 0 < e < 1/4. It is easy to get i ((ogn)a\p(\Sn \ > (e + Kn)yl2nlogn) -p(\N\>(e + Kn)^logn) \

= i ^g^\^\Sn\> (e + Kn)yl 2n log n) - p(\N \ > (e + Kn2logn) \

n<H(e) n

+ i > (e + Kn)^2nlogn)-P(\N\ > (e + Kn2logn) \ = h+12.

n>H(e)

We first consider I1. Let An = supx \P(\Sn\ > x-^/n) - P( | N\ > x) |, noting Theorem 1.1, since W(x) is a continuous fUnction, then, for any x > 0, we have limn-M An = 0. It follows

£2«+2Ii < £2a+2 £ a„ = £2«+2 £ ^^ A„

n<H(£) n n<H(£) n

< ^ .ZjoP A (3-9)

< C""1^ - 0, as e 1 0.

Note that (1/(logn)a+1) Zn<H(e)((logn)a/n)An - 0, e I 0, then (3.9) holds. Turnto I2,one can get

I2 < ^ M^N\>(e + Kn)^2\ogn)

n>H(e) n

(l )a (3.10)

+ ^ ^og«^p^ |Sn| > (e + ^2nlog«) ± I3+14.

n>H (e) n

Without loss of generality, we can assume that 0 < e < 1/4, M > 4, then one can get H(e) - 1 > VH(e). Note, in particular, that the definition of Kn, for n large enough, we have \Kn\ < e/4. Then, for I3, it follows that

e 13 < e

1 ^Pf \ N\ > 2

n>H(e)

. o il^rr^a

)H(e)-1 X

i(ogx)ap(\ N\ > ¡^dx

(logx)^ e

(3.11)

P(\N\ > -,¡2logx)dx

ijHm x v1 1 ^ g J

< C y2a+1P( \ N\ > y)dy — 0, as M — o,

uniformly with respect to 0 < e < 1/4. We finally estimate I4, by Lemma 3.2, which yields, for n large enough,

P( |Sn| > (e + k n)^2nlogn j < P^ 1 Sn 1 > ~^2nlogn^j

< nP\ |X11 > — J2nlogn l+2em( 1+ e2logn V 11 2m^ h ) \ 2mEX2

_ I5 +16,

(3.12)

where m is a positive integer to be specified later. Then, observe that n > H(e) implies log n > M/e2, for I5, if -1 < a < 1, the proof is similar to that of Lemma 3.2 [15]; if a> 1, applying Fubini's theorem, it turns out that

1 ii^ _ 1 (,og„,aP (|X,|> ifm^

n>H(e) n>H(e) \ /

- , J2MJ J2M(j +1)

<C X (logn)^Pl^r<|X1|<^-Mmjr-

n>H (e) j_n \

(3.13)

j>H(e) 2m 2m n_H(e)

< clj»*irP(j < 2m^ <j+l)- m

j>H(e)

CEXKlog |X1 0'

Furthermore, one can easily get limsupei0 e2a+2 ^n>H(e)(logn)aI5/n _ 0. We finally estimate I6, by the arbitrarity of m(> 1), one can obviously choose an appropriate positive

integer m, such that m > a +1. Then we have

y (logn)a c y (logn)a ( e2logn m , n 6 " n \2mEX2

n>H(e) n>H(e) \ 1

y (logn)a , 21 r

< c x —-— (£2logn> n

n>H (e)

,Ce-2m f- / (log

(3.14)

)H(e)-1 \ X

< Ce-2m(log(H(e)))a+1

< Ce-2a-2^a+1-m

it is easy to get e2a+2 Xn>H(e)(logn)aI6/n = 0, uniformly with respect to 0 < e <

1/4. Thus the proof of Proposition 3.3 is completed. □

Proof of Theorem 2.1. Combining Propositions 3.1 and 3.3, one can complete the proof ofthis theorem immediately. □

4. Proof of Theorem 2.4

The following propositions will simplify the proof of Theorem 2.4, which are stated as follows.

Proposition 4.1. Suppose that {W (t) : t > 0} be a standard Wiener process (Brownian motion). Then

lime2a+2 J (logp( sup | W(s) | > (e + K„)J2logn)

ei0 n=i n ^0<s<1 '

™ (-1)n = r«(a +1)-1E\N\2a+2 = (2n + 1)2a+2 •

Proof. Noting the result of Billingsley [18], p( sup | W(s)| > x) = 1 - X (-1)kP((2k - 1)x < N < (2k + 1)x)

= ^(-1)kP(N > (2k +1)x) = 2X(-1)kP(\N \ > (2k +1)x),

k=0 k=0

where N is the standard normal random variable. Then, according to Proposition 3.1, one can complete the proof easily. □

Lemma 4.2 [7]. Suppose that {Xn : n > 1} be strictly stationary NA sequences, EXi = ^,

=2 EXiXn > 0, set Sm

0 < VarX1 = a2 < o, and B2 = EX\ + 2 X = EX1Xn > 0, set Sm = £ = xk, write

Wn(t) = 77^ (Sm + (nt - m)Xm+1 - ntu), m < n<m + 1, 0 < t < T. (4.3) B^Jn

Wn(t) - W(t) in C[0, T], (4.4)

where W (t) is the standard Wiener process and C[0, T ] is the usual C space on [0, T ].

Lemma 4.3 [11]. Let {Xn : n > 1} be a sequence ofNA random variable with mean zero and finite second moments. Set Sn = X1 + ■■■ + Xn and Bf = X 1=1 EXf. Then for all x > 0, a > 0, and 0 < 1,

p( max |Sk| > x) < 2p( max I Xk I > a) + —^exp( -, n_, |. (4.5)

\1<k<nl k| / \1<k<« k| J 1 - fi \ 2(ax + Bf) J

Proposition 4.4. Suppose that EXf < o, if-1 <a < 1/2; EXf(log \X1\)a < oo, if a > 1/2. Then

lim e2a+2 X ((ognaiP(Mn > (e + kJJ 2n log n )

n>1 n (4.6)

- p( sup | W(s) | > (e + Kn)J2logn) I = 0.

0<s<1 ' 1

Proof. Let H(e) be as above, it follows that

X (lognn)" p(Mn > (e + Kn)yl 2n log n) - p( sup |W (s)| > (e + Kn2lognj

n=1 0<s<1

= X (log n) pM > (e + Kn)J 2n log n) - p( sup | W (s) | > (e + Kn)J 2logn)

n<H(e) n ^iss1 '

+ X (logn) P(Mn > (e + Kn)J2nlogn) - p( sup |W(s) | > (e + Kn)J2logn)

n 0 s 1

n>H(e)

= I1+12.

Noting Lemma 4.2, we have Mn/*Jn — sup0sts 1 | W(t) |, as n -»to. Similar to Theorem 2.1, one can get lime|0 e2a+2I[ = 0. We now estimate ¡2, it turns out that

¡2< X ^-^Pf sup | W (s) | > (e + Kn)J 2logn)

n>H(e) n Vo<s<J y

a (4.8)

+ X (log n)" P( max 1 Sk l> (e + Kn)j2n log n) = ¡3 + ¡4-

n>H(e) n Kl<k<n V ^

Observe that P(sup0<ss1 |W(s)| >x) < 2P(|N| > x), see [18]. Similar to Theorem 2.1, we have lime|0 e2a+2I'i = 0. We then consider ¡4, as a matter of fact, by Lemma 4.3, take x = eJ2nlogn/2, a = (2neJlogn)1/2. For n large enough, one could get

P max lSk 1 > (e + Kn) J2nlogn < P max 1 Sk 1 > — J2nlogn

1<k<n v ) \ 1<k<n 2 v

< 2nP ( l X11 > (2ne^jlognj 1 ) + exP

pe2log n

V 8( ^a/^)3/2 + 1)/

= ¡5+1'-

Without loss of generality, we can assume 0 < e < 1/4, M > 16, Notice that n > H(e) if and only if log n > M/e2, then for ¡6, we have

2«+2 X Ce2a+2 X (log n). exp ^ ^^ ^

n>H(e) n>H(e)

i I-) 1/2

^ 2a+2 f" „ w ejlogX) \ dx

< ^ J H(e)-1(logx)aexp {-9-] dx

f - / -p( eJogx)1/2 x dx (4-10)

< Ce2a+2 (logx)a exp ^ \g ' dx J-JW) \ 9 J x

< Ce2a+2\ e-2a-2y4a+3exp(-y)dy

Jp 4M/9

< C y4a+3 exp(-y)dy -- 0, as M ■

Jp 4M/9

We then estimate I'5 .If -1 < a < 1/2, the following result is obvious according to EX2 < to; if a> 1/2, by Fubini's theorem, it follows that

I = Z (log„)ap(|X1|* (2ne^g~n)1/2)

>H(e) n n>H (e)

< c i (logn)a 1 p(4k < Vk+r)

n>H (e) k=n \ V4M /

< C I p(Vk < vfc+T) 1 (logn)a (4.11)

k>H(e) \ V4M ' n=H(e)

< CX k(logk)ap(Vk < v^1)

,CFX2(log |X1 0 a

< CEX1 V4M <a

Proof of Theorem 2.4. The proof follows from Propositions 4.1 and 4.4. □

Acknowledgments

The author would like to express his deep thanks to the referee for careful reading and valuable suggestions. This work is supported by the Natural Science Foundation of Department of Education ofZhejiang Province (Grant no. 20060237).

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Yuexu Zhao: Institute of Applied Mathematics and Engineering Computation,

Hangzhou Dianzi University, Hangzhou 310018, China

Email address: yxzhao@hdu.edu.cn

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