On bounds for convergence rates in combinatorial strong limit theorems and its applications

We find necessary and sufficient conditions for convergences of series of weighted probabilities of large deviations for combinatorial sums $\sum_i X_{ni\pi_n(i)}$, where $||X_{nij}||$ is a matrix of order $n$ of independent random variables and $(\pi_n(1), \pi_n(2), \ldots , \pi_n(n))$ is a random permutation with the uniform distribution on the set of permutations of numbers $1, 2, \ldots , n$, independent with $X_{nij}$. We obtain combinatorial variants of results on convergence rates in the strong law of large numbers and the law of the iterated logarithm under conditions closed to optimal ones. We discuss applications to rank statistics.