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CHAPTER 3 Convergence of Random Variables 3.1 Introduct ion In probability and statistics, it is of ten ne cess ar y to co nsider the distribution of a random v ar iable that is itself a func ti on of several random variables, for example, Y = g(X 1 , ··· ,X n ); a simple example is the sample mean of random variables X 1 , ··· ,X n . 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 informa tio n about the joi nt distributi on of X 1 , ··· ,X n 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 X 1 , ··· ,X n 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 in fi nit e 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 approac h, of course, is ver y common in mathematics. For exa mple, if n is large compar ed to x, one might approxi mate (1 + x/n) n by exp(x) since lim n→∞ 1 + x n n = exp(x). (However, this approximation may be very poor if x/n is not close to 0.) A more interesting example is Stirling’s approximation, which is used to approximate n! for large values of n: n! ≈ √ 2π exp(−n)n n+1/2 = s(n) where the approximation holds in the sense that n!/s(n) → 1 as c 2000 by Chapman & Hall/CRC

[3] Convergence of Random Variables

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