Two Proofs of the Central Limit Theorem
نویسنده
چکیده
In this lecture, we describe two proofs of a central theorem of mathematics , namely the central limit theorem. One will be using cumulants, and the other using moments. Actually, our proofs won't be entirely formal, but we will explain how to make them formal. What it the central limit theorem? The theorem says that under rather general circumstances, if you sum independent random variables and normalize them accordingly, then at the limit (when you sum lots of them) you'll get a normal distribution. For reference, here is the density of the normal distribution N (µ, σ 2) with mean µ and variance σ 2 : 1 √ 2πσ 2 e − (x−µ) 2 2σ 2. We now state a very weak form of the central limit theorem. Suppose that X i are independent, identically distributed random variables with zero mean and variance σ 2. Then X 1 + · · · + X n √ n −→ N (0, σ 2). Note that if the variables do not have zero mean, we can always normalize them by subtracting the expectation from them. The meaning of Y n −→ Y is as follows: for each interval [a, b], Pr[a ≤ Y n ≤ b] −→ Pr[a ≤ Y ≤ b].
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