The Convergence For Some Random Sequences

In the present paper, let {Xn, n≥1} be a stationary sequence of random variableswith heavy tails. we study the logarithmic asymptotic behaviors for the distributionsof the partial sums S∑n=ni=1Xiunder the assumption that {Xn, n≥1} is a sequenceof dependent random variables. Our main interest is in the crude estimates P(|Sn|>nx)≈n αx+1for appropriate values x where α is a specifc parameter. Some resultsin this paper extend the works of Hu and Nyrhinen [13].In addition, we also study the simple linear errors in variables (EV) model: ηi=θ+βxi+εi, ξi=xi+δi, with i.i.d. errors (εi, δi),(1≤i≤n). The consistency andasymptotic normality for the LS estimatorsβ nandθ nof the unknown parameters β, θare obtained, which weaken some known conditions and improve some known results.Lastly, let {Xn, n≥1} be a sequence of m-dependent random variables withsurvival function F (x)=P{X1> x}. The empirical survival function Fn(x) basedon X1,···, Xnis proposed as an estimator for F (x). In this paper, we obtain someasymptotic results of the survival function estimator, which improve and extend theworks of Zarei in [29].