Precise Asymptotic Properties Of The Two Types Of Sequence Of Random Variables,

This thesis is finished during my Master of Science, it consists of two chapters.In Chapter â… , there are some results about the moving-average processes. We assume that {ε_i;—∞ < i < ∞} is a doubly infinite sequence of i.i.d random variables with mean zeros and finite variables. Let {α_i;—∞ < i < ∞} is an absolutely summable sequence of real numbers andLi (2005) discussed the precise asymptotics result in this kind of complete moment convergence for moving average process. Set S_n = , n ≥ 1. We reduce the moment condition of random variable sequence of {ε_i;— ∞ < i < ∞} and getTheorem 1.2.1 Suppose that {X_k;k ≥ 1} is defined as (1.1.1), where {α_i;-∞ < i < ∞} is 0 sequence of real numbers with and {ε_i;— ∞ < i < ∞} is a sequence of i.i.d random variables with E_ε1 = 0, E_ε1² < ∞. Then for 11+ p/2, if E|ε₁|^r <∞, we havewhere Z has a normal distribution with mean 0 and variance Ï„² = σ² .Moreover, for the condition of r = 2, p = 2 in the above theorem, we get the following result.Theorem 1.2.2 Suppose that {X_k;k ≥ 1} is defined as (1.1.1), where {α_i;-∞ < i < ∞} is a sequence of real numbers with and {ε_i;-∞ < i <∞} is a sequence of i.i.d random variables with E_ε1 = 0, E_ε1² < ∞. Let 0 ≤ δ ≤ 1, α be a positive number, and 1/2 - 1/α < δ < 1 - 1/α. Then we havewhere Z has a normal distribution with mean 0 and variance Ï„² = σ² .In Chapter â…¡, we study the largest entries of sample covariance matrix. In section 2, we discuss the complete convergence of the largest entries of sample covariance matrix. In section 3, we get the precise asymptotics results for it.Let Xn = (xij) be an n by p data matrix, where the n rows are observations from a certain multivariate distribution and each of p columns is an observation from a variable of the population distribution. Suppose that {ï¿¡, x^;i,j = 1,2,...} are i.i.d random variables with Eï¿¡ = 0,Varï¿¡ = 1. Let pij be the Pearson correlation coefficient between the ith and jth columns of Xn.That is- Xj)where X{ = (1/n) Y%=1 xk,i i then Rn := (p^) is a p by p symmetric matrix. It is called the sample correlation matrix generated by Xn.Jiang(2004) chose the intuitive one Ln = maxi 0. Ifn/p -> 7 € (0,oo), then for any q ï¿¡ R and e > 0 , we haven=300.Theorem 2.3.1 Suppose that Ejï¿¡|30+a < 00 for some a > 0. J/n/p -? 7 € (0,oo), then for any —1/2 < q < 0 , toeRemarks. Notice that when q = —1/2 , Theorem 2.3.1 doesn't hold. Instead, we getTheorem 2.3.2 Suppose that ï¿¡]f |30+a < 00 for some a>0. Ifn/p -4 7 e (0,00), thenwe have1 °° 1 lim , . ,—— Y I_<sub>P(Wn > (2 + e)y/nlogn') = K.