CHAPTER 3 Convergence of Random Variables

  • Published 2002

Abstract

In probability and statistics, it is often necessary to consider the distribution of a random variable that is itself a function of several random variables, for example, Y = g(X1, · · · , Xn); a simple example is the sample mean of random variables X1, · · · , Xn. Unfortunately, finding the distribution exactly is often very difficult or very time-consuming even if the joint distribution of the random variables is known exactly. In other cases, we may have only partial information about the joint distribution of X1, · · · , Xn in which case it is impossible to determine the distribution of Y . However, when n is large, it may be possible to obtain approximations to the distribution of Y even when only partial information about X1, · · · , Xn is available; in many cases, these approximations can be remarkably accurate. The standard approach to approximating a distribution function is to consider the distribution function as part of an infinite sequence of distribution functions; we then try to find a “limiting” distribution for the sequence and use that limiting distribution to approximate the distribution of the random variable in question. This approach, of course, is very common in mathematics. For example, if n is large compared to x, one might approximate (1 + x/n)n by exp(x) since

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@inproceedings{2002CHAPTER3C, title={CHAPTER 3 Convergence of Random Variables}, author={}, year={2002} }