Precise asymptotics of complete moment convergence on moving average

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 $X_k = sumnolimits_{i = - infty }^{ + infty } {a_i xi _{i + k} }$ be a moving average process. Under some proper moment conditions, the precise asymptotics are established for $$mathop {lim }limits_{varepsilon searrow 0} frac{1} {{ - log varepsilon }}sumlimits_{n = 1}^infty {frac{1} {{n^2 }}ES_n^2 Ileft{ {left| {S_n } right| geqslant nvarepsilon } right} = 2EZ^2 .}$$ where Z ～ N (0, τ ²), τ ² = σ ²(Σ _n=?∞ ^∞ a _i )² and $$mathop {lim }limits_{varepsilon searrow 0} varepsilon ^{2delta } sumlimits_{n = 2}^infty {frac{{(log n)^{delta - 1} }} {{n^2 }}ES_n^2 Ileft{ {left| {S_n } right| geqslant sqrt {nlog nvarepsilon } } right} = frac{{tau ^{2delta + 2} }} {delta }Eleft| N right|^{2delta + 2} .}$$