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On an arithmetical property of moments and cumulants

Ashot V. Kakosyan, Lev B. Klebanov

Abstract

The main result of the paper is the following. Let a non-degenerate distribution have finite moments $μ_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{μ_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms or at most three. In the latter case, this sequence has the form $\{1,a,1-b,a,1-b,a,1-b, \ldots\}$ and corresponds to a distribution with the characteristic function \begin{equation*}\label{ eq0} f(t)=\frac{1+iat+bt^2}{1+t^2}, \quad \text{where} \;\; b\geq 0,\; \frac{1-a-b}{2}\geq 0,\; \frac{1+a-b}{2}\geq 0. \end{equation*}.

On an arithmetical property of moments and cumulants

Abstract

The main result of the paper is the following. Let a non-degenerate distribution have finite moments of all orders . Then the sequence either contains infinitely many different terms or at most three. In the latter case, this sequence has the form and corresponds to a distribution with the characteristic function \begin{equation*}\label{ eq0} f(t)=\frac{1+iat+bt^2}{1+t^2}, \quad \text{where} \;\; b\geq 0,\; \frac{1-a-b}{2}\geq 0,\; \frac{1+a-b}{2}\geq 0. \end{equation*}.
Paper Structure (3 sections, 3 theorems, 26 equations)

This paper contains 3 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Property of moments
  3. Property of cumulants

Key Result

Theorem 1.1

Let a non-degenerate distribution have finite moments $\mu_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{\mu_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms or at most three. In the latter case, this sequence has the form $\{1,a,1-b,a,1-b,a,1-b, \ldots\}$ and cor .

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • proof