Large Deviations for a Terminating Compound Renewal Process

Let (ξ(i),η(i)) ∈ Rᵈ⁺¹, i ∈ N, be independent and identically distributed random vectors, let ξ(i)∈ Rᵈ be random vectors, let η(i) be improper nonnegative random variables, and let P(η(i) = +∞)∈(0,1). It is assumed that the distribution of the vector (ξ(1),η(1)) subject to \η(1)<+∞\ satisfies the Cramér condition. By a terminating compound renewal process we mean the process ZT =∑ₖ₌₁NTξ(k), where NT = max ∈ N η(1)++η(k) ≤ T\ is the renewal process corresponding to improper random variables η(1), η(2),. We find precise asymptotics of the probabilities P(ZT∈ IΔT(x)) and P(ZT = x) in the nonlattice and strongly arithmetic cases, respectively; here IΔT(x)=∈Rᵈ xⱼ≤ yⱼ < xⱼ+ΔT, j=1,,d\, and ΔT is a positive function converging sufficiently slowly to zero.