Precise asymptotics of complete moment convergence on moving average

Abstract

Let {ξ _i ,−∞ < i < ∞} be a doubly infinite sequence of identically distributed φ-mixing random variables with zero means and finite variances, {a _i ,−∞ < i < ∞} be an absolutely summable sequence of real numbers and $X_k = \sum\nolimits_{i = - \infty }^{ + \infty } {a_i \xi _{i + k} }$ be a moving average process. Under some proper moment conditions, the precise asymptotics are established for

$$\mathop {\lim }\limits_{\varepsilon \searrow 0} \frac{1} {{ - \log \varepsilon }}\sum\limits_{n = 1}^\infty {\frac{1} {{n^2 }}ES_n^2 I\left\{ {\left| {S_n } \right| \geqslant n\varepsilon } \right\} = 2EZ^2 .}$$

where Z ∼ N (0, τ ²), τ ² = σ ²(Σ ^∞_n=−∞ a _i )² and

$$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2\delta } \sum\limits_{n = 2}^\infty {\frac{{(\log n)^{\delta - 1} }} {{n^2 }}ES_n^2 I\left\{ {\left| {S_n } \right| \geqslant \sqrt {n\log n\varepsilon } } \right\} = \frac{{\tau ^{2\delta + 2} }} {\delta }E\left| N \right|^{2\delta + 2} .}$$