Asymptotic Properties Of The Random Product Of Trimmed Products And Some Limit Theorems Of NA Sequences

This thesis is finished during my Master of science and it consists of three chapters.Chapter I is on the asymptotic properties of the random product of trimmed products.As we all known, a sequence of independent identically distributed positive square integrable random variables, theirs products are asymptotically lognormal. This fact is an immediated consequence of the classical central limit theorem(clt). In the present paper,many scholars primarily research the limiting law of the products of sums of iid rvs. Arnold and Villasenor(1998) obtained the following version of the clt for a sequence (X_n) of i.i.d exponential rvs with the mean equal oneRempala and Wsolowski(2002) got the above theorem:Theorem A Let (X_n) be a sequence of iid positive square integrable rvs. Denote μ = E(X₁)>0, the coefficient of variation r =σ/μ, where σ² = Var(X₁), and S_k — X₁ + ?? ?+ X_k, k — 1,2, ? ? ? . Thenwhere N is a standard normal rv. And then they obtained another results in 2005Theorem B Let (X_k,ii=1, ??? ,k; k = 1,2, ??? be a triangular array of iid positive square integrable rvs with finite absolute moment of order p > 2. Denote μ = E(X₁) > 0,r= σ/μ, where σ² = Var(X₁), and S_k = X(k,i) +???X_k,k k = 1, 2, ??? . Then as n→∞where N is a standard normal rv.In chapter I, under the condition of rvs owing the medium tail, I get the asymptotic property of the random product of the trimmed sum Tn(a):Theorem 1.1 Let (X_n) be a sequence of iid positive square integrable rvs. Denote μ= E(X₁) > 0, the coefficient of variation r = σ /μ, where σ² = Var(X₁), and S_k = X₁ + ... + X_k, k =1,2,_<sub><sub>Ï„_n are integral random variables,satisfying is a integral random variable.Then...