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 Lp norm λ,p of the order λ−δ/2 for all 1 p ∞, in the central limit theorem for a standardized random sum (SNλ − ESNλ)/ √ DSNλ , where SNλ = X1 + · · · + XNλ is the random sum of independent identically distributed random variables X,X1,X2, . . . with β2+δ = E|X|2+δ < ∞ where 0 < δ 1, Nλ is a random variable distributed by the Poisson distribution with the parameter λ > 0, and Nλ is independent of X1,X2, . . . .


Introduction
Let X, X 1 , X 2 , . . . be independent identically distributed random variables (r.v.'s) with µ = EX, α 2 = EX 2 , σ 2 = DX, and Let N λ be a r.v. distributed by the Poisson distribution with the parameter λ > 0 (for short, N λ ∼ P(λ)), i.e., Moreover, assume that the r.v. N λ is independent of the X 1 , X 2 , . . . , and consider the so-called Poisson random sum Here and in what follows R is a real line. We are interested in the rate of convergence of the L p norm λ,p for all 1 p ∞. First of all, note that there is a rich literature on normal approximations for random sums of independent sequences of r.v.'s, see, for example, papers [4,9], books [7,11], and the references therein. In most of these works the upper bounds of the uniform distance λ,∞ are investigated. The obtained bounds contain, as a rule, several terms, outlined by the distributions of summands X 1 , X 2 , . . . (not necessarily identically distributed r.v.'s), and the random numbers of summands N are also not necessarily Poissonian. These general bounds are provided with some loss of accuracy for concrete distribution of the random number of summands N .
Recall that Robbins (1948) [16] has proved that, if a distribution of a r.v. N depends on the parameter λ, the r.v. N has positive finite variance, i.e., 0 < DN < ∞, and then for a random sum S N = X 1 + · · · + X N , where X 1 , X 2 , . . . are independent identically distributed r.v.'s and N is independent of X 1 , X 2 , . . . , the following equality is valid In [20], we have obtained the upper estimates of L p norms for all 1 p ∞ of the order (DN) −1/2 in the limit (3) as N is either a Poisson r.v. or gamma one.
In this paper, we have obtained the upper estimate of the L p norm λ,p for all 1 p ∞ (see Theorem 1). The obtained constants are not the best possible, but that was not the main author's aim.
The uniform estimate (5) of λ,∞ in the case δ = 1 was first presented in [15], and the proof independently was given in [3] and [1]. It has been proved independently in [14] and [10] that the constant C ∞ (1) in (5) in this particular case (p = ∞, δ = 1) is the same as in the Berry-Esséen inequality for the sum of a non-random number of independent identically distributed summands (C ∞ (1) 0.7655 [17]).
As in [20], to obtain the upper estimates of the norm λ,∞ (uniform metric) and the L 1 norm λ,1 , we have formed linear differential equations from the characteristic function of the standardized r.v.
by virtue of which we succeeded in getting proper estimates of differences: between the characteristic function and the normal one, and between their derivatives as well. The proof of the estimate of the L p norm is elementary. The papers, for example, [5] and [19], in which the rate of convergence of L p norms in the central limit theorem is investigated for sums of independent random variables, are close to that of ours.
The obtained results are produced under the influence of Stein's method and the papers of Stein [18] and Tikhomirov [21].
Denote the characteristic function of the standardized r.v.

Ee itZ N λ , and the derivative of the characteristic function f λ (t) with respect to t by f λ (t).
To prove Theorem 1, we use an auxiliary result, Lemma 1, on the behaviour of the functions f λ (t) and f λ (t).

LEMMA 1. Let the conditions of Theorem 1 be satisfied. Then the characteristic function f λ (t) satisfies the following homogeneous linear differential equation for all t ∈ R:
Moreover, for all |t| C 0 (δ) (8) where C 0 (δ) = (1+δ)(2+δ) Proof. It is easy to see that ES N λ = λµ, DS N λ = λ(µ 2 + σ 2 ) = λα 2 , and To make sure on the correctness of (6), it suffices to take the derivatives with respect to t on both sides of expression (9). Denote by θ 1 and θ 2 complex functions such that |θ i | 1.
Theorem 1 is proved.