Theorems on large deviations for the sum of random number of summands

Abstract

In this paper, we present the rate of convergence of normal approximation and the theorem on large deviations for a compound process _{Zt = \sumNt i=1 t aiXi,} where Z₀ = 0 and a_i > 0, of weighted independent identically distributed random variables X_i, i = 1, 2, . . . with  mean EX_i = µ and variance DX_i = σ² > 0. It is assumed that N_t is a non-negative integervalued random variable, which depends on t > 0 and is independent of X_i, i = 1, 2, . . . .