Summary
For independent identically distributed bivariate random vectors (X 1, Y 1), (X 2, Y 2), ... and for large t the distribution of X 1 +...+ X N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y 1, Y 2,.... Similar expansions are derived for multivariate X 1. Furthermore, local asymptotic expansions are valid for the distribution of f(X 1)+ ...+ f(X N ) when N is large and nonrandom, and X i , i=1, 2,..., is a discrete strongly mixing Markov chain.
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Hipp, C. Asymptotic expansions in the central limit theorem for compound and Markov processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 361–385 (1985). https://doi.org/10.1007/BF00532740
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DOI: https://doi.org/10.1007/BF00532740