Strong Law Of Large Numbers Of Independent Sequences And The Sequence Of Pairwise Nqd Law,

For sequences of independent identically distributed random variables(i.i.d.r.v.), there are two classic strong laws of large numbers (SLLN), whichare Kolmogorov SLLN and Marcinkiewicz SLLN. Etemadi, Martikainen,Petrov and so on have improved the results. But the SLLN for i.i.d.r.v, areneeded to be generalized further.About the SLLN for sequences of NA random variables, people have re-searched much. Almost all the results for sequences of i.i.d.r.v, hold for se-quences of NA random variables. But the SLLN for pairwise NQD randomvariables aren't so rich, and general results are even fewer.This thesis is mainly about the strong laws of large numbers for se-quences of independent identically distributed random variables (i.i.d.r.v.) andsequences of pairwise NQD random variables. It is composed by two chapters.The main results of each chapter are listed below:Chapter 1. The strong law of large numbers for sequences ofi.i.d.r.v.'sIn this chapter, two general strong laws of large numbers for sequences ofi.i.d.r.v, are presented. As a corollary, the Marcinkiewicz SLLN is given. Themain theorems are as follows:Hypothesis A:Let f(x),g(x) be positive real functions defined on the same domain[h,+∞),φ(x)=f(x)g(x), (0≤h≤1. f(x) or g(x) may not be well de-fined at the point h, but if so, (?) f(x)g(x) exists, and letφ(h) equal to thelimit at this point.) and they satisfy the following conditions:(1) g(n)≠0(n∈N);(2) (?) f(x)=+∞, and f(n) is increasing (n∈N);(3)φ(x) is strictly increasing on [h,+∞),(?)φ(x)=+∞, and its range is (4) There exists a constant c, such that for everym≥2, integral from n=m-1 to +∞dx/φ²(x)≤c integral from n=m to +∞dx/φ²(x);(5) There exist constants a, b∈R, such that for every t∈R, t² integral from n=φ^-1(|t|) to +∞(dx)/(φ²(x))≤αφ^-1(|t|)+b.Theorem 1.1 (A general SLLN for i.i.d.r.v, sequences)Let f(x),g(x),φ(x) befunctions satisfying the conditions in HypothesisA. Let {X_n,n∈N} be a sequence of i.i.d.r.v., and Y is a positive randomvariable with finite expectation, let L_n=EX₁I_{[|X1|≤φ(n)]},σ_n=1/(f(n)) sum from n=k=1 to n (X_k-L_k)/(g(k)).(1) Ifσ_nâ†'0 a.s. and |X_n|≤Y a.s. then E[φ^-1(|X₁|)]＜∞.(2) If E[φ^-1(|X₁|)]＜+∞, thenσ_nâ†'0 a.s.Theorem 1.2 (further result than the former)Let f(x),g(x),φ(x)=f(x)g(x) be functions satisfying the conditions inHypothesis A, andφ(x) satisfies the following conditions:(â…°) If integral from n=r to +∞dx/φ(x)is finite, then integral from n=r to +∞dx/φ(x)≤lr/φ(r) (r≥1, l is aconstant);(â…±) If integral from n=r to +∞dx/φ(x) doesn't exist or is infinity, then (?) x/φ(x)=+∞andintegral from n=1 to t dx/φ(x)≤mt/φ(t) (r≥1,t≥1, m is a Constant).Let {X_n} be a sequence of i.i.d.r.v., and Y is a positive random variablewith finite expectation, and Let L_n=EX₁I_{[|X1|＜φ(n)]},σ_n=1/(f(n)) sum from n=k=1 to n (X_k-L_k)/(g(k)),σ_n¹=1/(f(n)) sum from n=k=1 to n (X_k)/(g(k)).(1) Ifσ_nâ†'0 a.s., and |X_n|≤Y a.s. then E|φ^-1(|X₁|)]＜+∞. (2) If E[φ^-1(|X₁|)]＜+∞(when (â…±) holds, EX₁=0), thenσ_n¹â†'0 a.s.Chapter 2. The strong law of large numbers for sequences ofpairwise NQD random variables.The main aim of this chapter is to extend the results of chapter 1 tosequences of pairwise NQD random variables. And as corollaries, MarcinkiewiczSLLN and another SLLN for sequences of pairwise NQD random variables arepresented. The main theorems are as follows:Theorem 2.2 (A general SLLN for pairwise NQD sequences)Let f(x),g(x),φ(x) satisfy the conditions in Hypothesis A. let {X_n} bea sequence of pairwise NQD identically distributed random variables, Y is apositive random variable with finite expectation, and let log be the logarithmwith any positive real number as its base, L_n=E(-φ(n)I_{[X1＜-φ(n)]}+X₁I_{[|X1|≤φ(n)]}+φ(n)I_{[X1＞φ(n)]}),σ_n=1/(f(n)) sum from n=k=1 to n (X_k-L_k)/(g(k)), (?)_n=1/(f(n)log n) sum from n=k=1 to n (X_k-L_k)/(g(k)).(1) Ifσ_nâ†'0 a.s. and |X_n|≤Y a.s. then E[φ^-1(|X₁|)]＜+∞.(2) If E[φ^-1(|X₁|)]＜+∞, then (?)_nâ†'0 a.s.Theorem 2.3 (further result than theorem 2.2)Let f(x), g(x),φ₁(x)=f(x)g(x) satisfy the requirements of Hypothesis A,andφ₂(x)=f(x)g(x)log x satisfies the requirements of Theorem 1.2. let {X_n}be a sequence of pairwise NQD identically distributed random variables, andY is a positive random variable with finite expectation, and let L_n,σ_n be thesame as those in Theorem 2.2. Letσ_n¹=1/(f(n)log n) sum from n=k=1 to n (X_k)/(g(k)),(1) Ifσ_nâ†'0 a.s., and |X_n|≤Y a.s. the E[φ^-1(|X₁|)]＜+∞.(2) If E[φ^-1(|X₁|)]＜+∞(whenφ₂(x) satisfies condition (â…±) of Theorem1.2, EX₁=0), thenσ_n¹â†'0 a.s.