Law of large numbers vs central limit theorem

Question

The law of large numbers and the central limit theorem are both statements about the sample mean of i.i.d. random variables.

Explain precisely what each theorem says, how they differ, and why neither implies that a run of losses must be "balanced out" by future wins.

Accepted Answer

Let X1, X2, \dots be i.i.d. with mean \mu (and, for the CLT, finite variance \sigma^2). LLN (where the mean goes). The law of large numbers says \bar{X}n 	o \mu, in probability (weak LLN) or almost surely (strong LLN). It pins down the limit but says nothing about how fast, or about the shape of the fluctuations. CLT (how it fluctuates on the way). The central limit theorem refines this: the error \bar{X}n - \mu is of size \sigma/\sqrt{n} and, once rescaled, \sqrt{n}\,(\bar{X}n - \mu) \xrightarrow{d} \mathcal{N}(0, \sigma^2). So the LLN is a zeroth-order statement (the fluctuation dies), whi…

Law of large numbers vs central limit theorem

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