Random Summation

Limit Theorems and Applications

Boris V. Gnedenko, Victor Yu. Korolev

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000141177
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

This book provides an introduction to the asymptotic theory of random summation, combining a strict exposition of the foundations of this theory and recent results. It also includes a description of its applications to solving practical problems in hardware and software reliability, insurance, finance, and more. The authors show how practice interacts with theory, and how new mathematical formulations of problems appear and develop. Attention is mainly focused on transfer theorems, description of the classes of limit laws, and criteria for convergence of distributions of sums for a random number of random variables. Theoretical background is given for the choice of approximations for the distribution of stock prices or surplus processes. General mathematical theory of reliability growth of modified systems, including software, is presented. Special sections deal with doubling with repair, rarefaction of renewal processes, limit theorems for supercritical Galton-Watson processes, information properties of probability distributions, and asymptotic behavior of doubly stochastic Poisson processes. Random Summation: Limit Theorems and Applications will be of use to specialists and students in probability theory, mathematical statistics, and stochastic processes, as well as to financial mathematicians, actuaries, and to engineers desiring to improve probability models for solving practical problems and for finding new approaches to the construction of mathematical models.

Chapter 1

Examples

1.1 Examples related to generalized Poisson laws

EXAMPLE 1.1.1. Consider the energy received by some region of the surface of Earth from cosmic particles during time T. Let E_k be the energy of the k-th particle and N be the total number of particles arrived during time T. It is obvious that the energy received by the region is equal to

$E=E_1+\dots +E_N$

(1.1)

We will consider E_k, k ≥ 1, here, as independent random variables with the same distribution function F(x), and N as a random variable which is independent of the variables {E_k}_k≥1, and takes nonnegative integer values with the corresponding probabilities

$p_n=\text{P}\left(N=n\right),n=0,1,2,\dots$

When studying these physical phenomena it is usually assumed that the random variable N has the Poisson distribution with some parameter λ, i.e.,

$p_n=\frac{{\text{λ}}^n}{n!}{e}^{-\text{λ}},n=0,1,2,\dots ,\text{λ}>0$

(1.2)

Experimental data are in good accordance with this assumption. Some more details of the mathematical premises of (12) are presented in the Appendix, where the properties of the Poisson flows of events are described.

Let us find the distribution function G(x) of the random variable E. Since the sum (11) has no summands with probability p₀, then we can conclude that the function G(x) has a jump at zero, which is equal to p₀. The sum (11) will have n summands with probability p_n and the distribution function of the variable E will be F^*n(x), where

$F^{*}{}^{\left(n+1\right)}\left(x\right)={\displaystyle {\int }_0^x{F}^{*n}\left(x-z\right)dF\left(z\right)}$

and F^*1(x) = F(x). If we denote by F^*(0)(x) the degenerate distribution function which has the only unit jump at zero, then according to the law of total probability we can write the following relation

$G\left(x\right)=\text{P}\left(E<x\right)={\displaystyle \sum _{\text{n=0}}^{\infty }{p}_n{F}^{*n}\left(x\right)}$

(1.3)

In terms of the characteristic functions

$g\left(t\right)={\displaystyle {\int }_0^{\infty }{e}^{itx}dG\left(x\right)f\left(t\right)={\displaystyle {\int }_0^{\infty }{e}^{itx}dF\left(x\right)}}$

relation (13) can be written as

$g\left(t\right)={\displaystyle \sum _{k=0}^{\infty }{p}_k{f}^k\left(t\right).}$

(1.4)

If p_k are given by (12), then (14) takes a simple form

$g\left(t\right)=\exp \left\{\text{λ}\left[f\left(t\right)-1\right]\right\}.$

(1.5)

Formula (15) implies, in particular, that the distribution function G(x) is always infinitely divisible (including the case when the distribution function F(x) is not).

It is well known that under the conditions of our example the expectation and variance of the random variable E are determined as

$\begin{array}{l}\text{E}E=\text{E}N\cdot \text{E}{E}_1,\hfill \\ \text{D}E=\text{E}N\cdot \text{D}{E}_1+\text{D}N{\left(\text{E}{E}_1\right)}^2.\hfill \end{array}$

Under assumption (12) these formulas take a very simple form

$\text{E}E=\text{λ}\cdot \text{E}{E}_1,\text{D}E=\text{λ}\cdot \text{E}{E}_1^2.$

This example seemed to be an interesting and useful object of the mathematical investigation. Due to these reasons the formula for the expectation of a random sum (when the number of summands is independent of the summands) was included into the textbook (Gnedenko, 1949). We should note that other authors paid their at...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Table of Contents
• 1 Examples
• 2 Doubling with Repair
• 3 Limit Theorems for “Growing” Random Sums
• 4 Limit Theorems for Random Sums in the Double Array Scheme
• 5 Mathematical Theory of Reliability Growth. A Bayesian Approach
• Appendix 1. Information Properties of Probability Distributions
• Appendix 2. Asymptotic Behavior of Generalized Doubly Stochastic Poisson Processes
• Bibliographical Commentary
• References
• Index