PRECISE RATES IN THE LAW OF ITERATED LOGARITHM FOR THE MOMENT CONVERGENCE OF φ-MIXING SEQUENCES

Let {X-n vertical bar n >= 1} be a strictly stationary sequence of phi-mixing random variables with zero mean and finite variance. Set S-n = Sigma(n)(i=1) X-i and M-n = max(1 <= k < n) vertical bar S-k vertical bar, n >= 1. For d > 0 and a(n) = o((log log n)(-d)), we show the exact rates in the law of iterated logarithm of a kind of weighted infinite series of E{vertical bar S-n vertical bar - (epsilon + a(n))sigma root n(log log n)(d)}(+) and E{M-n - (epsilon + a(n))sigma root n(log log n)(d)}(+) as epsilon SE arrow 0. (C) 2015 Mathematical Institute Slovak Academy of Sciences