Local Theorems for Arithmetic Multidimensional Compound Renewal Processes under Cramér’s Condition

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Abstract

We continue the study of compound renewal processes (c.r.p.) under Cramér’smoment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensionalc.r.p. Z(n) and Y(n), for which the random vector ξ = (τ, ζ) controlling these processes (τ > 0 defines the distance between jumps, ζ defines the value of jumps of the c.r.p.)has an arithmetic distribution and satisfies Cramér’s moment condition. For theseprocesses, we find the exact asymptotics in the local limit theorems for the probabilities P (Z(n) = x), P (Y(n) = x) in theCramér zone of deviations for x ∈ Zd (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p.,where the vector ξ = (τ, ζ) has a nonlattice distribution).

Original languageEnglish
Pages (from-to)284-302
Number of pages19
JournalSiberian Advances in Mathematics
Volume30
Issue number4
DOIs
Publication statusPublished - Nov 2020

Keywords

  • arithmetic distribution
  • compound renewal process
  • Cramér’s condition
  • deviations function
  • large deviations
  • local limit theorem
  • moderate large deviations
  • renewal function

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