Large deviations of generalized renewal process

Let ( ξ ( i ), η ( i )) ∈ ℝ d +1 , 1 ≤ i < ∞, be independent identically distributed random vectors, η ( i ) be nonnegative random variables, the vector ( ξ (1), η (1)) satisfy the Cramer condition. On the base of renewal process, N T = max{ k : η (1) + … + η ( k ) ≤ T } we define the generalized renewal process Z T = array ∑ᵢ₌₁NT array ξ ( i ). Put I Δ T ( x ) = { y ∈ ℝ d : x j ≤ y j < x j + Δ T , j = 1, …, d }. We find asymptotic formulas for the probabilities P ( Z T ∈ I Δ T ( x )) as Δ T → 0 and P ( Z T = x ) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of ( ξ (1), η (1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.