Spectral methods – central limit theorem
نویسنده
چکیده
Now we recall the statement of the central limit theorem (CLT) and give a proof in the case of IID (independent identically distributed) random variables. The weak law of large numbers says that if Xn is a sequence of IID random variables with E[Xn] = 0, then writing Sn = ∑n−1 k=0 Xk, the time averages 1 n Sn converge to 0 in probability, or equivalently (since the limit is a constant), in distribution. In the case when σ = E[X] < ∞, the central limit theorem strengthens this to the result that the sequence 1 √ n Sn converges in distribution to N(0, σ), the normal distribution with mean 0 and variance σ. That is, we have
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