On the rate of convergence of Lp norms in the CLT for Poisson random sum

Abstract

In the paper, we present the upper bound of L_p norm \delta_λ,p of the order λ^-δ/2 for all 1 \leq  p \leq ∞,  in the central limit theorem for a standardized random sum (S_Nλ - ES_Nλ)/DS_Nλ , where S_Nλ = X₁ + ··· + X_Nλ is the random sum of independent identically distributed random variables X, X₁, X₂, . . . with  β_2+δ = E|X|^2+δ < ∞ where 0 < δ \leq 1, N_λ is a random variable distributed by the Poisson distribution with the parameter λ > 0, and N_λ is independent of X₁, X₂, . . ..