On longest matching consecutive subsequence

The longest matching consecutive subsequence between two [Formula: see text]-ary expansions found applications in information theory and molecular biology. In this paper, we establish an almost sure limit theorem for the longest matching consecutive subsequence, and investigate the size of the set of points violating the limit in terms of Hausdorff dimension.

Almost sure limit theorem for the order statistics of stationary Gaussian sequences

In this paper, by using a new comparison inequality for order statistics of Gaussian variables, we proved an almost sure central limit theorem for extreme order statistics of stationary Gaussian sequences with covariance rn under the condition rn log n(log log n)1+? = O(1) for some ? > 0. A similar result on intermediate order statistics is also proved for stationary Gaussian sequences. The obtained results improve some of the existing results.

Almost Sure Central Limit Theory for Self-Normalized Products of Sums of Partial Sums

LetX,X1,X2,…be a sequence of independent and identically distributed random variables in the domain of attraction of a normal law. An almost sure limit theorem for the self-normalized products of sums of partial sums is established.