1.5 The central limit theorem

Let $\{X_l\}$ denote a sequence of independent, identically distributed, random variables. Assuming they havezero means and finite variances (equaling $^2$ ), the Central Limit Theorem states that the sum $\sum_{l=1}^L \frac{X_l}{\sqrt{L}}$ converges in distribution to a Gaussian random variable. $$\frac{1}{\sqrt{L}}\sum_{l=1}^L X_l\stackrel{L}{}(0, ^2)$$ Because of its generality, this theorem is often used to simplify calculations involving finite sums of non-Gaussian random variables. However, attention is seldompaid to the convergence rate of the Central Limit Theorem. Kolmogorov, the famous twentieth centurymathematician, is reputed to have said, "The Central Limit Theorem is a dangerous tool in the hands of amateurs." Let'ssee what he meant.

Taking $^2=1$ , the key result is that the magnitude of the difference between $P(x)$ , defined to be the probability that the sum given above exceeds $x$ , and $Q(x)$ , the probability that a unit-variance Gaussian random variable exceeds $x$ , is bounded by a quantity inversely related to the square root of $L$ ( Cramer: Theorem 24 ). $$\left|P(x)-Q(x)\right|\le c\frac{(\left|X\right|^3)}{^3}\frac{1}{\sqrt{L}}$$ The constant of proportionality $c$ is a number known to be about 0.8 ( Hall: p6 ). The ratio of absolute third moment of $X_l$ to the cube of its standard deviation, known as the skew and denoted by ${}_X$ , depends only on the distribution of $X_l$ and is independent of scale. This bound on the absolute error has been shown to be tight ( Cramer: pp. 79ff ). Using our lower bound for $Q()$ (see ), we find that the relative error in the Central Limit Theoremapproximation to the distribution of finite sums is bounded for $x> 0$ as