Asymptotic expansions in the central limit theorem for compound and Markov processes

Summary For independent identically distributed bivariate random vectors (X₁, Y₁), (X₂, Y₂), ... and for large t the distribution of X₁ +...+ X_N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y₁, Y₂,.... Similar expansions are derived for multivariate X₁. Furthermore, local asymptotic expansions are valid for the distribution of f(X₁)+ ...+ f(X_N) when N is large and nonrandom, and X_i, i=1, 2,..., is a discrete strongly mixing Markov chain.