Limit Theorems For Dependent Sequences Of Random Variables

Probability limit theory is not only one of the main branches of probability theory, but also is an important theoretical foundation of other fields of probability theory and mathematical statistics. The famous probabilist Kolmogorov from previous Soviet Union said: " Only probability limit theory can reveal the epistemological value of probability. Without it, you couldn't understand the real meaning of the fundamental conceptions in probability." The classical central limit theorem is an essential foundation of probability theory. It is extensively applied to statistics, nature sciences, engineering and economics. Its methods and results continue to have great influence on other fields of probability theory, mathematical statistics, and their applications. The almost sure central limit theorem and self-normalized limit theory have become two important fields of the study of probability limit theory in recent decades. Some profound results of almost sure central limit theorem and self-normalized limit theorem have been reached through deep research in this dissertation.Let {X_n,n ≥ 1} be a sequence of independent and identically distributed random variables. Brosamler (1988) and Schatte (1988) first independently discovered the almost sure central limit theorem (ASCLT). The result is as follows: If {X_n,n ≥ 1} is a sequence of independent and identically distributed random variables with mean 0 and variance 1, and E|X_i|^2+δ < ∞ (δ > 0). Then for all xWhere I is an indicator function. From then on, many authors have studied the almost sure central limit theorem. Lacey and Philipp (1990), Schatte (1991), Cso|¨go|″ and Horváth (1992), Berkes and Dehling (1994), Berkes (1995) obtained the almost sure central limit theorem of independent and identically distributed random variables. Berkes and Dehling (1993), Berkes and Csáki (2001) considered the almost sure central limit theorem of independent but not identically distributed random variables. Peligrad and Shao (1995) and Lesigne (1999) discussed the case of dependent random variables. The general form of ASCLT follows a common pattern that for a sequence of random variables with partial sums satisfyingfor some constant sequences {On}, {&?} and some random variable Y with nondegenerate distribution function G, then under certain conditions, we have1 n 1 lim-----y^-IiSi-ai^ =G(x) a.s.for any continuity point x of G. Fahrner and Stadtmiiller (1998), Cheng et al. (1998), Fahrner (2000) and Stadtmiiller (2002) proved almost sure limit theorems for maxima of independent and identically distributed random variables. Berkes and Csaki (2001) got almost sure limit theorems for nonlinear functionals of independent random variables. Their results immediately imply analogue of ASCLT for partial sums, almost sure limit theorems for partial maxima, extreme order statistics, empirical distribution functions, [/-statistics, local times and invariance principle.Since we always meet with the dependent random variables in the applications of probability theory and mathematical statistics, it is necessary to study the almost sure limit theorems of dependent random variables. In section 2 of chapter 1, we obtain almost sure limit theorems for maxima of a stationary sequence of normal random variables under weaker conditions:Let {Xn, n > 1} be a stationary standardized normal sequence with covariance rn — Cov(X-L,Xn+i), satisfying rn —* 0 as n —> oo, andfor some e > 0, 7 > 2, C > 0. Put Mn = maxi 1 andpn\ogn<(loglogn)i+￡-Let the constants {Uni\ be such that X)iLi(l—$(um)) is bounded and Xn = mini c(logn)1//2 for some c > 0. If 52!￡=i(l $(uni)) —? r as n —*? oo for some t > 0, thenJiSL tIZZ 1^ rJink (f<,L, u = exp(-T) a.s.(a) Let {￡n,n > 1} be a nonstationary standardized Gaussian sequence with covariance ry = Cov(^,￡j) and satisfy the following conditions:(1) S = sup^ |ro-| < 1.(2) For some 7 > \^' and some positive number d, we havefor some positive numbers A and e.Let On = (2 log n) 2 and 6n = (2 log n) 2 — (log log ft + log 4?r)/(2(2 log n) 2), a;is a real number. ThenIn section 4 of chapter 1, we further study the almost sure limit theorem of maxima of random variables and give some almost sure limit theorems for maxima of a triangular array of normal random variables, we obtain: Let {Xn,i}, n = 1,2,.... i = 0,1,2,... be a triangular array of normal random variables, such that for each n, {Xnii,i > 0} is a stationary normal sequence. Under some conditions, for all x e (-00,00),1 " 1 lim :-----5^ tIs v <■ < \\ = exp(—1? exp(—a;)) a.s.In section 2 of chapter 2, we firstly extend the result of Berkes and Csaki (2001) about independent random variables and give a general result of almost sure limit theorem for a sequence of random elements in a complete separable metric space. Let {Yn,n > 1} be a sequence of random elements in B. Suppose that there exist a nondecreasing sequence of positive numbers {Cn} with limn-HjoCn, = oo,cri+i/cn = 0(1) and B-valued random elements Yk,i, fe, I e N, k < I, such that for any function g ￡ BL(B) and k < I(^)IXfor some C > 0, e > 0. Let {<4, k > 1} satisfy 0 < 4 < log(cfc+i/cfc) and ￡jg.j dk = oo and set Z?n = 2k=i d*. Then for any probability distribution /x on the o-algebra B of B^ kyk M ^.s. as n -+ ooif and only if1 n^ta'*/* asn-^oo.^n fc=i Where => denotes weak convergence.And then we apply it to a stationary sequence of associated (negatively associated), strongly mixing, p-mixing, Bernoulli shift and Gaussian random variables.In section 3 of chapter 2, we discuss the almost sure central limit theorems for functions of random variables. One of the main results is as follows: Let {Xn,n > 1} be a sequence of centered random variables and /fc(Xi, ? ? ■, Xk) be a random function satisfyingh(Xu--,Xk) = aSk + br^Xr, ■ ■ ■, Xk),where a,b € R,a ^ 0, supnE|rn| < oo and rn = o{an). Assume that for any function 9 € BL(R)If |a =4> 7V(0,1) as n —> oo, then1 n 1 lim-----Y^ yljju^ , = $(z) a.s.In section 4 of chapter 2, we study the almost sure limit theorems for the product of sums of random variables under a new weak dependence condition. Let {Xn,n > 1} be a stationary a-weakly dependent sequence of positive random variables with EXi = fj,\ and VarXi = of < oo. Denote 7 = limn-,00 VarSV^n = a2 > 0 and VarSn,j- - VarS^-i > ^ for constants 0, then forany real a;1 "1 lim Y\I F1()where F{x) is the distribution function of the random variable e^.In section 5 of diapter 2, we get ASCLT for functionals of absolutely regular processes. In section 6 of chapter 2, we prove some almost sure functional limit theorems for empirical processes.In chapter 3, we discuss the self-normalized limit theorem for a sequence of weakly dependent random variables. Put Sn = ￡"=i X<, V% = E"=i Xf, n > 1. It is well-known that moment conditions or other related conditions are necessary and sufficient for many classical limit theorems. For example, the strong law of large numbers holds if and only if the mean of X is finite;the central limit theorem holds if and only if EX2/{|X| < x} is slowing varying as x —> oo. On the other hand, limit theorems for self-normalized sums Sn/Vn put a totally new countenance upon the classical limit theorems. In contrast to the well-known Hartman-Wintner law of the iterated logarithm and its converse by Strassen (1966), Griffin and Kuelbs (1989) obtained a self-normalized law of the iterated logarithm for all distributions in the domain of attraction of a normal law or stable law. Shao (1997) Showed that no moment conditions are needed for a self-normalized large deviation result P(Sn/Vn > rr-Vn), while Gine, Gotze and Mason (1997) proved that the tails of Sn/Vn are uniformly sub-Gaussian when the sequence is stochastically bounded. Jing et al. (2003) established a Cramer type result for self-normalized sums only under a finite (2 + S)th moment. Chistyakov and Gotze (2004) obtained the sufficient and necessary conditions for Sn/Vn converge to a random variable. These results show that self-normalized limit theorems are valid without any moment condition or under little moment condition and the results are much more neater. More importantly, self-normalization is more nature from the statistical point of view because the parameters involved in many calssical limit theorems are usually unknown, one has to use some statistics to estimate them first. A typical case is the Student ^-statistic Tn. The close relationship between the Student t—statistic Tn and the self-normalized sum Sn/Vn can be seen below:n-1 xi/2andIn section 2 of chapter 3, we get self-normalized central limit theorem for weakly dependent random variables. The main result is as follows: Let {Xn, n> 1} be a stationary sequence of (0, C\, ^)-weakly dependent random variables. ￡X\ = 0, ￡X\ < oo. If limn-.oo Var(Sn)/n = a2 > 0, for some c € [0,2], d > 0,, u, v) = {u + v)d{Lip(h) + Lip{k))c,and for some D > (d - c/2) V 0, 9T = O(r^D), thenThe intention of improvement of various conditions lies in extending the applied scope of theorems. It makes them to apply to various sequences of dependent random variables. It is difficult to build probability inequality of satisfying the SLLN of logarithm. And the difference between the proof of the former and mine lies in applying the comparison inequality and some technical inference. For example, the conditions of Theorem 2.2.1 are more general and the proof is more concise.we obtained the sufficient and necessary conditions for some results in this dissertation, for example, the results about the almost sure limit theorem for a sequence of random elements in a complete separable metric space in the section 2.2 and the results about the almost sure limit theorems for maxima of a triangular array of normal random variables, the conditions in these results can not be weaken any more. Though the author tried his best to make each of the results as perfect as possible, a few results are still not satisfied, for example, the conditions of Theorem 1.3.1 in the section 1.3, the applications of Theorem 2.2.3 in the section 2.2 and the self-normalized limit theorem for dependent random variables are needed to study further.This dissertation is made up of some papers which were written by author in the past three years. Some of these papers have been published and accepted, and the rest have been submitted to various journals about probability theory. Details are attached to the appendix. Due to my limited knowledge, errors may incur in this dissertation, so your criticism would be greatly appreciated.