We quantify the elementary Borel–Cantelli Lemma by higher moments of overlap count statistic in terms weighted summability probabilities. Applications include mean deviation frequencies Strong Law and Iterated Logarithm.

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes · · ·X−1, X0, X1, · · · whose partial sums Sn = X1 + · · · + Xn are of the form Sn = Mn+Rn, where Mn is a square integrable martingale with stationary increments and Rn is a remainder term for which E(R 2 n) = o(n). Here we explore the Law of the Iterated Logarithm (LIL) for the...