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Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings

Fraser Daly

Abstract

Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable random variables, where the random variable $N$ is independent of the $X_j$, and the $X_j$ are from the generalized multinomial model introduced by Tallis (1962). This relaxes the classical assumption that the $X_j$ are independent. We use zero-biased coupling and its generalizations to give explicit error bounds in the approximation of $Y$ by a Gaussian random variable in Wasserstein distance when either the random variables $X_j$ are centred or $N$ has a Poisson distribution. We further establish an explicit bound for the approximation of $Y$ by a gamma distribution in stop-loss distance for the special case where $N$ is Poisson. Finally, we briefly comment on analogous Poisson approximation results that make use of size-biased couplings. The special case of independent $X_j$ is given special attention throughout. As well as establishing results which extend beyond the independent setting, our bounds are shown to be competitive with known results in the independent case.

Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings

Abstract

Let be a sum of a random number of exchangeable random variables, where the random variable is independent of the , and the are from the generalized multinomial model introduced by Tallis (1962). This relaxes the classical assumption that the are independent. We use zero-biased coupling and its generalizations to give explicit error bounds in the approximation of by a Gaussian random variable in Wasserstein distance when either the random variables are centred or has a Poisson distribution. We further establish an explicit bound for the approximation of by a gamma distribution in stop-loss distance for the special case where is Poisson. Finally, we briefly comment on analogous Poisson approximation results that make use of size-biased couplings. The special case of independent is given special attention throughout. As well as establishing results which extend beyond the independent setting, our bounds are shown to be competitive with known results in the independent case.

Paper Structure

This paper contains 9 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Size biasing
  3. Zero biasing
  4. Gaussian approximation
  5. The zero-mean case
  6. The independent case
  7. The Poisson case
  8. Gamma approximation in the Poisson case
  9. Poisson approximation

Key Result

Lemma 2.1

Let $N$ be a non-negative, integer-valued random variable, and let $X_1,X_2,\ldots$ be random variables with mean zero which satisfy (eq:model) and are independent of $N$. Let $Y=X_1+\cdots+X_N$ and $Y^{z}$ denote its zero-biased version. Let and let $I_\tau$ be a Bernoulli random variable, independent of all else, with $\mathbb{P}(I_\tau=1)=1-\mathbb{P}(I_\tau=0)=\tau$. Then where $N^s$ is the

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • ...and 7 more