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Central Limit Theorem Definition
Central Limit Theorem (CLT) states that given certain conditions then the mean of a sufficiently large number of statistical independence random variables each with finite mean and variance, will be approximately normal distribution and the central limit theorem has a number of variants. In its simplest form, the random variables must be identically distributed and in variants, convergence of the mean to the normal distribution also occurs for non-identical distributions given that they follow certain conditions.
In more general Probability theory, a central limit theorem is a set of weak convergence of measure theories as they express the fact that a sum of many independent and identically distributed (i.i.d) random variable or alternatively random variables with specific types of dependence, will tend to be distributed. This is according to one of a small set of attractor distributions. If the variance of the i.i.d. variables are finite then the attractor distribution is a normal distribution and in contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 and therefore having infinite variance will tend to an alpha-stable distribution with stability parameter or index of stability of α as the number of variables grows.
The Central Limit Theorem for independent process assume that {X1, ..., Xn} be a random sample of size n which is a sequence of independent and identically distributed random variables drawn from distributions of expected value which is given by µ and finite variance which is given by σ2 . Then the sample mean of the random variables are represented as follow
The theorem can be stated as follows suppose {X1, X2, ...} is a sequence of Independent and identically distributed random variables with E[Xi] = µ and Var[Xi] = σ2 < ∞ . Then as n approaches infinity, the random variables √n(Sn − µ) Convergence in distribution to a normal distribution which is represented as follows N(0, σ2)
In the case σ > 0, convergence in distribution means that the cumulative distribution function of √n(Sn − µ) converge point wise to the cdf of the N(0, σ2) distribution: for every real number z it is represented as follows:
where Φ(x) is the standard normal cdf that is calculated at x and the convergence is uniform in z in the sense that where sup is the least upper bound.
