1-Dependent Stationary Sequences for Some Given Joint Distributions of Two Consecutive Random Variables

We provide a method to construct a 1-dependent stationary sequence provided some mixing condition on the joint distribution of two consecutive random variables. Two illustrations of computational benefits of the method are given. We obtain analytical formulas to compute the expectation and variance of the number of occurrences of a word in a sequence of letters from a finite alphabet generated by the 1-dependent model.We also obtain an approximation formula for the distribution of the longest success run in a Bernoulli sequence generated by our model.