Limit theorems for the number of dense series in a random sequence

We investigate the joint distribution of the number of dense series in a random sequence over a finite alphabet. With the use of the Chen–Stein method, we find estimates of the distance in variation between the distribution of the vector of the numbers of dense series of ones of given lengths and the accompanying multidimensional Poisson distribution. These estimates give a possibility to prove limit theorems of Poisson type for the numbers of dense series of ones of given lengths and of lengths no less than a given length, for the number of intervals densely filled by ones, a limit theorem for the maximal length of dense series of ones, and a limit theorem for the number of dense series of ones of a given weight.