The central limit theorem (CLT) is a fundamental theorem in probability theory that states that the sum of a large number of independent, identically distributed random variables (i.e., a sample) will be approximately normally distributed, regardless of the underlying distribution of the individual random variables. The CLT has important applications in statistics and many other fields, and it underpins much of modern statistical theory.

There are several different versions of the CLT, each with slightly different assumptions and hypotheses. Here, I will present one of the most commonly used versions, known as the Lindeberg-Levy CLT.

Suppose we have a sample of n independent, identically distributed random variables X1, X2, ..., Xn, with mean μ and variance σ^2. Let Y_n be the sample mean of these random variables, i.e.,

Y_n = (X1 + X2 + ... + Xn) / n

Then, the Lindeberg-Levy CLT states that as n approaches infinity, the distribution of Y_n approaches a normal distribution with mean μ and variance σ^2/n, i.e.,

Y_n ~ N(μ, σ^2/n)

In other words, the larger the sample size, the more closely the sample mean will approximate a normal distribution. This approximation is typically quite good, even for relatively small sample sizes (e.g., n = 30 or more).

The proof of the Lindeberg-Levy CLT is based on the characteristic function of the sample mean Y_n, which is defined as:

φ_n(t) = E[e^(itY_n)]

Using the properties of the characteristic function and some algebraic manipulation, it can be shown that as n approaches infinity, φ_n(t) converges pointwise to the characteristic function of a normal distribution with mean μ and variance σ^2/n, i.e.,

φ_n(t) → e^(iμt - σ^2t^2/2n)

This result is known as the central limit theorem for characteristic functions, and it implies that the distribution of Y_n approaches a normal distribution as n approaches infinity, as stated above.

The Lindeberg-Levy CLT is a powerful and widely used result in statistics and other fields, and it has many important applications. However, it is important to note that the CLT only holds under certain conditions and assumptions, and these must be checked carefully in practice to ensure that the results are valid.
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