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Probability, Fourth Edition by Rick Durrett

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3Central Limit Theorems

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.

3.1 The De Moivre-Laplace Theorem

Let X1, X2, . . . be i.i.d. with P(X1 = 1) = P(X1 = −1) = 1/2 and let Sn = X1 + · · · + Xn. In words, we are betting $1 on the flipping of a fair coin and Sn is our winnings at time n. If n and k are integers

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since S2n = 2k if and only if there are n + k flips that are +1 and nk flips that are − 1 in the first 2n. The first factor gives the ...

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