Precise asymptotics of complete moment convergence on moving average

Let {xi(i),-infinity < i < infinity} be a doubly infinite sequence of identically distributed phi-mixing random variables with zero means and finite variances, {a(i),-infinity < i < infinity} be an absolutely summable sequence of real numbers and X-k = Sigma(+infinity)(i=-infinity) a(i)xi(i+k) be a moving average process. Under some proper moment conditions, the precise asymptotics are established for (lim)(epsilon SE arrow 0)1/-log epsilon Sigma(infinity)(n=1)1/n(2)ES(n)(2)I{vertical bar S-n vertical bar >= n epsilon}=2EZ(2). where Z similar to N(0,tau(2)), tau(2) = sigma(2) (Sigma(infinity)(i=-infinity) a(i))(2), and (lim)(epsilon SE arrow 0) epsilon(2 delta) Sigma(infinity)(n=2)(log n)(delta-1)/n(2)ES(n)(2)I{vertical bar S-n vertical bar >=root n log n epsilon} = tau(2 delta+2)/delta E vertical bar N vertical bar(2 delta+2).