Strong Convergence Of Dependent Random Variables

This thesis is finished during my master of science, mainly discusses SLLN andcomplete convergence for dependent random variables. It consist of three chapters:In chapterâ… , we mainly discuss the strong law of large numbers and completeconvergence for pairwise PQD sequences.The concept of pairwise PQD sequences was introduced by Lehmann. It is a kindof generic sequences, including associated sequences. Matula established a almost ev-erywhere central limit theorem; Yan Jigao proved a strong convergence for Jamison-wise weighted sums of pairwise PQD sequences; Lu Fengbin obtained a completeconvergence which is similar to PA sequences. In this chapter, the following resultsare obtained:Theorem0.1 Suppose{X_n; n≥1}is a pairwise PQD sequence with mean 0, andsum from i=1 to∞v^1/2(2ⁱ)＜∞. Let 1≤p＜2;φ: Râ†'R⁺ is nonnegative, even and continuousfunction, (?)φ(x)=∞, for some 1＜s≤2,φ(x)/x↑andφ(x)/x⁸↓, xâ†'∞. Assume sum from n=1 to∞log² n[(?) Eφ(|X_i|)/φ(n^1/p]^1/2＜∞Then 1/n^1/p sum from i=1 to n X_iâ†'0, a.s. nâ†'∞.Theorem0.2 Suppose{X_n; n≥1} is a pairwise PQD sequence, for(?)ε＞0, sum from j=1 to∞VarX_j/j²+sum from j≠k=1 to∞Cov(X_j, K_k)/j·k＜∞and sum from j≠k=1 to∞P(|sum from i=1 to j(X_i-EX_i)|≥jε, |sum from i=1 to k(X_i-EX_i)|≥kε)＜∞.Then sum from n=1 to∞(logn)^-2P(|sum from i=1 to n(X_i-EX_i)|≥ε)＜∞.In chapterâ…¡, we mainly discussed the SLLN for a sequence of nonmonotonic func-tions of associated random variables.The following definition was given by Newman in 1984:Definition0.1 Let f and f₁ be two real-valued functions defined on Rⁿ, then f＜＜f₁ if and only if f+f₁, f-f₁ are both nondecreasing componentwise. In particular, if f＜＜f₁, then f₁ will be nondecreasing componentwise.Let{X_n; n≥1} be a PA sequences. Let(â…°) Y_n=f_n(X₁, X₂,…, X_n)(â…±) (?)_n=(?)_n(X₁, X₂,…, X_n) (0.0.4)(â…²) f_n＜＜(?)_n(â…³) EY_n²ï¼œâˆž, E(?)_n²ï¼œâˆž, n∈NIf the conditions (â…°)-(â…³) hold, we write Y_n＜＜(?)_n. There are some limiting resultson{Y_n; n≥1}. Matula(2001) proved SLLN and CLT for {Y_n; n≥1}, Dewan andRao(2006)obtained Hajek-Renyi-tpye inequlity. We prove the following result:Theorem0.3 {X_n; n≥1} is a PA sequence, Y_n, (?)_n is defined in (0.0.1), andY_n＜＜(?)_n. g_n(X) is even functions, and is positive and nondecreasing when x＞0, forevery n satisfies the alternative assumption that:(â…°) x/g_n(x) is nondecreasing in (0,∞);(â…±) x/g_n(x), g_n(x)/x² are nonincreasing in (0,∞). Meanwhile EY_n=E(?)_n=0.In addition, {a_n; n≥1} is a sequence of real numbers, with 0＜a_n↑∞, sum from n=1 to∞Eg_n(Y_n)/g_n(a_n)＜∞and sum from n=1 to∞(Eg_n((?)_n))/g_n(a_n)^1/2＜∞. Then when nâ†'∞, 1/a_n sum from k=1 to n Y_kâ†'0 a.s.In addition, we prove a complete convergence for {Y_n; n≥1} similar to Theo-rem0.2.In chapterâ…¢, a complete convergence for linear processes under dependent as-sumption is discussed.Assume that {X_i; -∞＜i＜∞} is a doubly infinite sequence, let {a_i; -∞＜i＜∞} be an absolutely summable sequence of real numbers, and {Y_k; k≥1}: Y_k=sum from i=-∞to∞a_i+kX_i. (0.0.5)Linear processes defined as (0.0.5) are called moving average processes. Manylimiting results were obtained for moving average processes {Y_k; k≥1}. Burton andDehling(1990) obtained large deviation principle assuming Eexp(tX₁)＜∞; Ibragimov(1962) established CLT; Li et al.(1992) obtained a complete convergence. In sectionâ…¡, weprove a complete convergence when{X_i; -∞＜i＜∞} is a pairwise NQD sequence:Theorem0.5 Supposel＜p＜2, 1/2＜α＜1,αp≥1. Let(â…°) h(x)＞0(x＞0) be a slowly varying function, when xâ†'∞;(â…±) {a_i; -∞＜i＜∞} be an absolutely summable sequence of real numbers;(â…²) EX₁=0, EX₁²ï¼œâˆž. ThenE|X₁|^ph(|X₁^1/α|)＜∞imply sum from n=1 to∞n^αp-2h(n)P(|sum from k=1 to n Y_k|≥n^αε)＜∞, (?)ε＞0.Let{Y_t; t∈Z⁺} be a linear process defined on a probability space (Ω, (?), P): Y_t=sum from j=0 to∞a_jX_t-j. (0.0.6)where {a_j; j≥0} is a sequence of real numbers, sum from j=0 to∞|a_j|＜∞; {X_t; t∈Z⁺} is a processand EX_t=0, 0＜EX_t²ï¼œâˆž.Linear processes defined as (0.0.6) are called general linear processes. Obviously, moving average processes are special cases of general linear processes. The linear pro-cesses are of special importance in time series analysis. Fakhre Zaberi and Lee obtainedCLT under i.i.d assumption; in 1997, they obtained FCLT under the strong mixingcondition; Tae-Sun Kim, Mi-Hwa Ko and Dong Ho Park proved SLLN under theLPQD and PA condition on {X_t; t∈Z⁺}. In sectionâ…¢, we discuss the linear pro-cess under positive dependence condition on {X_t; t∈Z⁺}, and obtain the followingresult:Therorem0.6 Supposel＜p＜2, 1/2＜α≤1,αp≥1. Let(â…°) h(x)＞0(x＞0) be a slowly varying function, when xâ†'∞;(â…±) {a_j; j≥0} be a sequence of real numbers and sum from j=0 to∞|a_j|＜∞;(â…²) {X_n; n≥1} is a PA sequence bounded by X₀, EX_n =0 and sum from i=1 to∞v^1/2(2ⁱ)＜∞.Then E|X₀|^Ph(|X₀^1/α|)＜∞imply sum from n=1 to∞n^αp-2h(n)P(|S_n|≥εn^α)＜∞, (?)ε＞0.Meanwhile, we obtain a result under pairwise PQD condition on {X_t; t∈Z⁺}similar to the above.