On combinatorial strong law of large numbers and rank statistics

The author had earlier obtained a strong law of large numbers for combinatorial sums $\sum_iX_{ni\pi_n(i)}$, where $||X_{nij}||$ is a matrix of order $n$ from random variables with finite fourth moments and $(\pi_n(1), \pi_n(2), \ldots , \pi_n(n))$ is a random permutation having the uniform distribution on the set of all permutations of numbers $1, 2, \ldots , n$ and being independent from random variables $X_{nij}$ . The mutual independence for entries of the matrix has not been assumed. In the present paper, we derive the combinatorial SLLN under more general assumptions and discuss the behaviour of rank statistics.