On Estimation of Tail End Probabilities of the Sample Mean for Linear Stochastic Processes

Abstract

Let $\{X_j, j \geqq 1\}$ be a linear process defined by the relation $X_j = \sum^\infty_{r=0} g_v Y_{j-v}$, where $\{Y_j; j = 0, \pm 1, \cdots\}$ is a sequence of i.i.d. random variables which possess a $\operatorname{mgf} M_Y(t)$ over an open interval $I = (-c,c) (c > 0)$. Let $a$ be a fixed positive constant and denote the $\operatorname{mgf}$ of $Z_1 = Y_1 - a$ by $M_z(t)$. Assume that $E\{Y_1\} = 0, |g_v| \leqq C\rho_0^v (v \geqq 0)$ for some finite positive constants $\rho_0 (< 1), C$ and $\sum^\infty_{v = 0} g_v \neq 0$ (we can take $\sum^\infty_{v = 0} g_v = 1$ without loss of generality). Further assume that there exists a constant $\tau \in I_0, I_0 = (-cA^{-1}, cA^{-1}), A = \sum^\infty_{v=0}|g_v|$, such that $\rho = M_z(\tau) = \inf_{t \in I} M_z(t) < 1$ and $M_z'(\tau) = 0$. Then it is proved that for each $u = 0,1,2, \cdots$ we can find a bounded sequence $\{\beta_{u,n}\}$ of constants such that for any integer $r \geqq 3 P(\sum^n_{j=1}X_j/n \geqq a) = (\rho^n \lambda_n/\tau\sigma_n(2_\pi)^{\frac{1}{2}})\lbrack\sum^{r-3}_{u=0} \beta_{u,n} \sigma_n^{-u} + O(\sigma_n^{-(r - 2)})\rbrack$ as $n \rightarrow \infty$, where $\{\lambda_n\}$ and $\{\sigma_n\}$ are sequences of positive constants, and, as $n \rightarrow \infty, \lambda_n$ is bounded away from 0 and $\infty$ and $n^{-1}\sigma_n^2$ approaches a finite positive constant.