On the number of coincidences of two homogeneous random walks with positive increments

We investigate the distribution of the random variable equal to the number of coincidences of two homogeneous random walks with positive independent increments. This random variable is the length of the subsequence of common elements in two random sequences which are random subsequences of the same random sequence. For the considered random variable we obtain the asymptotic expression for the mathematical expectation and a limit theorem under the assumption that the sequential intervals between coincidences of the two random walks have a finite variance. For the particular case of random walks with increments equal to 1 and 2 we prove a finiteness of this variance and obtain the expression of the variance in terms of the parameters of the random walks.