An almost sure central limit theorem for the weight function sequences of NA random variables

Transcription

1 Proc. ndian Acad. Sci. (Math. Sci.) Vol. 2, No. 3, August 20, pp c ndian Academy of Sciences An almost sure central it theorem for the weight function sequences of NA rom variables QUNYNG WU College of Science, Guilin University of Technology, Guilin 54004, People s Republic of China wqy666@glite.edu.cn MS received January 200 Abstract. Consider the weight function sequences of NA rom variables. This paper proves that the almost sure central it theorem holds for the weight function sequences of NA rom variables. Our results generalize improve those on the almost sure central it theorem previously obtained from the i.i.d. case to NA sequences. Keywords. Weight function sequence; NA rom variables; almost sure central it theorem.. ntroduction main results DEFNTON Rom variables X, X 2,...,X n, n 2 are said to be negatively associated (NA) if for every pair of disjoint subsets A A 2 of {, 2,...,n}, cov( f (X i ; i A ), f 2 (X j ; j A 2 )) 0, where f f 2 are increasing for every variable (or decreasing for every variable) such that this covariance exists. A sequence of rom variables {X i ; i } is said to be NA if every finite subfamily is NA. Obviously, if {X i ; i } is a sequence of NA rom variables, { f i ; i } is a sequence of nondecreasing (or non-increasing) functions, then { f i (X i ); i } is also a sequence of NA rom variables. This definition was introduced by Joag-Dev Proschan [6]. Statistical test depends greatly on sampling. The rom sampling without replacement from a finite population is NA, but is not independent. NA sampling has wide applications such as in multivariate statistical analysis reliability theory. Because of the wide applications of NA sampling, the it behaviors of NA rom variables have received more more attention recently. One can refer to Joag-Dev Proschan [6] for fundamental properties, Matula [8] for the three series theorem, Shao [2] for the moment inequalities, Wu [4] for the strong consistency, Wu Jiang [5,6] for the law of the iterated logarithm. 369

2 370 Qunying Wu Starting with Brosamler [2] Schatte [], in the last two decades several authors investigated the almost sure central it theorem (ASCLT) for i.i.d. rom variables. We refer the reader to [2], [], [7], [5], [4], [9] [7]. The simplest form of the ASCLT [2,,7] reads as follows: Let {X n ; n } be i.i.d. rom variables with mean 0, variance σ 2 > 0 partial sums S n. Then ln n σ < x = (x) a.s. for any x R, (.) where denotes indicator function, (x) is the stard normal distribution function. Relation (.) is a logarithmic means of the sequence { S σ < x}; by Theorem of [], the arithmetic means of this sequence does not converge to (x) with probabity one, i.e. n σ < x = (x) a.s. for any x R (.2) fails. t is natural to as that if there exist other different averaging methods which also wor in the ASCLT, i.e., whether there exists a weight sequence {d = /; } such that d σ < x = (x) a.s. for any x R (.3) holds, where = n d. The terminology of summation procedures (see e.g. [3]) shows that the larger the weight sequence {D ; } in (.3) is, the stronger the relation becomes. By this argument, one should also expect to get stronger results if we use larger weights. And it would be of considerable interest to determine the optimal weights. The main purpose of this paper is to study function sequences of NA rom variables try to find some larger weight sequences {d ; } such that (.3) holds. n the following, a n b n denotes a n /b n, n, a n b n denotes that there exists a constant c > 0 such that a n cb n for sufficiently large n. The symbol C G denotes the set of continuity points of G, c sts for a generic positive constant which may differ from one place to another. Let c n > 0 with c n, c n+ = c <. (.4) c n f : R R ( =, 2,...) f,l : R l R ( < l) are increasing for every variable (or decreasing for every variable). Theorem. Let {X n ; n } be a sequence of NA rom variables satisfying E f l (X,...,X l ) f,l (X +,...,X l ) (c l /c ) δ (.5) Cov( f (X,...,X ), f,l (X +,...,X l )) (c l /c ) δ (.6)

3 Weight function sequences of NA rom variables 37 for some constant δ>0. Put d = ln c + c exp(ln α c ), 0 α</2, = Then for any distribution function G the relations d. (.7) d { f (X,...,X )<x} =G(x) a.s. for any x C G (.8) d P{ f (X,...,X )<x} =G(x) for any x C G (.9) are equivalent. n particular, let f (X,...,X ) = S σ.wehave Theorem 2. Let {X n ; n } be a stationary sequence of NA rom variables with EX = 0, 0 < EX 2 < σ 2 = EX =2 EX X > 0. Assume that there exists a constant γ>0 such that (c l /c ) γ l/, < l. (.0) Then d σ < x = (x) a.s. for any x R, (.) where d are defined by (.7). Remar. By the terminology of summation procedures, Theorems 2 remain valid if we replace the weight sequence {d ; } by {d ; } such that 0 d d, d =. Remar 2. Assume that {X n ; n } is a sequence of i.i.d. rom variables. () f c = α = 0, (.7) gives d e elnn. Thus (.) becomes (.). (2) Let c = in (.7), then d exp(ln α ). Forβ >, 0 <α</2, since d ˆ= log β d, Dn = (β + ) log β+ n. By Remar, (.) remains valid if we replace the weight sequence {d ; } by {d ; }. Thus our Theorem 2 not only generalize improve those on ASCLT previously obtained by Brosamler [2], Schatte [] Lacey Philipp [7] from the i.i.d. case to NA sequences but also exp the scope of the weights {d ; }.

4 372 Qunying Wu Remar 3. n (.7), let α =, c =, then d, n. Thus, by (.2) Remar, Theorem 2 does not hold for α. Whether Theorem 2 holds also for some /2 α< remains open. Remar 4. Equation (.0) is essential, if removed, Theorem 2 may not be set up. For example, let c = 2, we have d. Hence, Theorem 2 does not hold for c = 2. Remar 5. Let α>0 in (.7). By the Stolz theorem, (.4), ln(+x) x, e x x, for x 0 we get α ln α c n exp(ln α c n ). (.2) Remar 6. The weights in Theorems 2 have a very wide range. Taing different values of c α in Theorem 2, we can get a series of results such as the following corollary. COROLLARY Let {X n ; n } be a stationary sequence of NA rom variables with E X = 0, 0 < EX 2 < σ 2 = EX =2 EX X > 0. () f c =, (.7) gives d e, elnnforα = 0 d exp(ln α ), α ln α n exp(ln α n) for α>0. Thus (.) α exp(ln α ) ln α n exp(ln α n) σ < x = (x) a.s. for any x R hold. (2) Set lg 0 x = x denote lg j x = ln{max(e, lg j x)} for j. We also define lg x = lg x. f c = lg j, we have c l /c = (lg j l)/(lg j ) l/ by(lg j )/. ( Equation (.7) gives d ) j, i= lg i Dn lg j+ nforα = 0 d exp((lg j+ )α ) j, i= lg i α (lg j+ n) α exp((lg j+ n) α ) for α>0. Thus lg j+ n j σ < x = (x) lg i i= α (lg j+ n) α exp((lg j+ n) α ) exp((lg j+ ) α ) j lg i i= a.s. for any x R, σ < x = (x) a.s. for any x R.

5 Weight function sequences of NA rom variables 373 (3) f c = exp(ln β+ ), <β<0, we have c /c l /l by(exp(ln β+ ))/, (.7) gives d (β+) lnβ, lnβ+ n β+ for α = 0 d (β+) lnβ ln α(β+), α (ln n)(β+)( α) exp((ln n) α(β+) ) for α>0. Thus (β + ) 2 ln β+ n ln β α(β + ) (ln n) (β+)( α) exp((ln n) α(β+) ) σ < x = (x) a.s. for any x R ln β ln α(β+) σ < x = (x) a.s. for any x R. 2. Proofs Proof of Theorem. Denote by A the class of bound functions with bounded continuous derivatives. By Theorem 7. of [] 2 of [0], (.8) (.9) are equivalent to d g( f (X,...,X )) = g(x)dg(x) a.s. d Eg( f (X,...,X )) = g(x)dg(x) for every g A, respectively. Hence, it suffices to prove that for any g A, d (g( f (X,...,X )) Eg( f (X,...,X ))) = 0 a.s. (2.) Put ξ = g( f (X,...,X )) Eg( f (X,...,X )). Clearly, there is a constant c > 0 such that g(x) c, g (x) c, g(x) g(y) c x y for any x, y R, ξ 2c for any. (2.2)

6 374 Qunying Wu For any < l, note that f (X,...,X ) f,l (X +,...,X l ) are increasing for every variable (or decreasing for every variable), thus f (X,...,X ) f,l (X +,...,X l ) are NA. We get, using (.5), (.6), (2.2) Lemma 2.3 of [8], Eξ ξ l = Cov(g( f (X,...,X )), g( f l (X,...,X l ))) Cov(g( f (X,...,X )), g( f l (X,...,X l )) g( f,l (X +,...,X l ))) + Cov(g( f (X,...,X )), g( f,l (X +,...,X l ))) E f l (X,...,X l ) f,l (X +,...,X l ) + Cov( f (X,...,X ), f,l (X +,...,X l )) (c l /c ) δ (2.3) ( ) 2 E d ξ 2 d d l Eξ ξ l l n = 2 +2 l n;c l /c ln 2/δ d d l Eξ ξ l l n;c l /c <ln 2/δ d d l Eξ ξ l = 2(T n + T n2 ). (2.4) By (2.3), T n l n;c l /c ln 2/δ On the other h, by (.2), exp(ln α α c n ) (ln ) ( α)/α. Thus combining ξ 2cfor any, T n2 d d d l ln 2 D2 n ln 2. (2.5) l n;c l /c <(ln ) 2/δ (ln c l+ ln c l ) exp(ln α c l ) exp(ln α c n ) ln ln D2 n ln ln (ln ) ( α)/α. d

7 Weight function sequences of NA rom variables 375 Since α</2 implies ( 2α)/(2α) > 0 ε = /(2α) > 0. Thus, for sufficiently large n, we get T n2 D 2 n (ln ) /(2α) ln ln (ln ) ( 2α)/(2α) D 2 n (ln ) /(2α) = D 2 n (ln ) +ε. (2.6) Let T n = d ξ. Combining (2.4) (2.6), for sufficiently large n, we get ET 2 n (ln ) +ε. By (.4) (.7), we have +.Let0<η< +ε ε, n = inf{n; exp( η )}, then exp( η ), < exp( η ). Therefore i.e., exp( η ) exp( η ) <, exp( η ). Since ( η)( + ε) > from the definition of η, thus for any ε > 0, we have P( T n >ε ) ETn 2 <. ( η)(+ε) By the Borel Cantelli lemma, T n 0 a.s. Now for n < n n +,by ξ 2c for any, T n T n + 2c i=n + ( d i T n +2c D ) n 0 + a.s. + exp((+) η ) exp( η ) from = exp( η (( + /) η )) exp(( η) η ), i.e. (2.) holds. This completes the proof of Theorem. Proof of Theorem 2. Under the assumption of Theorem 2, by Corollary 2.2 of [3] we get i.e., S n σ n D N(0, ), P Sn σ n < x = (x) for any x R.

8 376 Qunying Wu Hence d P σ < x = (x) for any x R. By Theorem, to prove (.) it suffices to prove that (.5) (.6) hold for f l (X,...,X l ) = S l σ f l,l (X +,...,X l ) = S l S σ. By (.0), l E S l σ l S l S σ l = E S σ l EX 2 l (ES2 )/2 σ l ( ) l /2 ( ) γ/2 cl. c Andby0< m=2 EX X m = EX2 σ 2 2 < (.0), ( Cov σ, S ) l S σ = l σ 2 l Cov X i, = σ 2 l σ 2 l i= l i+ i= m= i+2 EX X m m=2 l j=+ X j Cov(X, X m ) ( ) l /2 This completes the proof of Theorem 2. ( cl c ) γ/2. Acnowledgements This wor was supported by the National Natural Science Foundation of China (0602), the support program of the New Century Guangxi, China, Ten-hundredthous Talents Project (200524), the support program of the Guangxi, China Science Foundation (200GXNSFA0320). References [] Billingsley P, Convergence of probability measures (New Yor: Wiley) (968) [2] Brosamler G A, An almost everywhere central it theorem, Math. Proc. Cambridge Philos. Soc. 04 (988) [3] Chraseharan K Minashisundaram S, Typical means (Oxford: Oxford University Press) (952)

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