The first four sections of this chapter develop the central limit theorem. The last five treat various extensions and complements. We begin this chapter by considering special cases of these results that can be treated by elementary computations.

Let *X*_{1}, *X*_{2}, . . . be i.i.d. with *P*(*X*_{1} = 1) = *P*(*X*_{1} = −1) = 1/2 and let *S*_{n} = *X*_{1} + · · · + *X*_{n}. In words, we are betting $1 on the flipping of a fair coin and *S*_{n} is our winnings at time *n*. If *n* and *k* are integers

since *S*_{2n} = 2*k* if and only if there are *n* + *k* flips that are +1 and *n* − *k* flips that are − 1 in the first 2*n*. The first factor gives the ...

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