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Concept# Asymptotic distribution

Summary

In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators.
A sequence of distributions corresponds to a sequence of random variables Zi for i = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of Zi converges to a probability distribution (the asymptotic distribution) as i increases: see convergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables is always zero or Zi = 0 as i approaches infinity. Here the asymptotic distribution is a degenerate distribution, corresponding to the value zero.
However, the most usual sense in which the term asymptotic distribution is used arises where the random variables Zi are modified by two sequences of non-random values. Thus if
converges in distribution to a non-degenerate distribution for two sequences {ai} and {bi} then Zi is said to have that distribution as its asymptotic distribution. If the distribution function of the asymptotic distribution is F then, for large n, the following approximations hold
If an asymptotic distribution exists, it is not necessarily true that any one outcome of the sequence of random variables is a convergent sequence of numbers. It is the sequence of probability distributions that converges.
Central limit theorem
Perhaps the most common distribution to arise as an asymptotic distribution is the normal distribution. In particular, the central limit theorem provides an example where the asymptotic distribution is the normal distribution.
Central limit theorem
Suppose is a sequence of i.i.d. random variables with and . Let be the average of . Then as approaches infinity, the random variables converge in distribution to a normal :
The central limit theorem gives only an asymptotic distribution.

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Asymptotic distribution

In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators. A sequence of distributions corresponds to a sequence of random variables Zi for i = 1, 2, ..., I .

Central limit theorem

In probability theory, the central limit theorem (CLT) establishes that, in many situations, for independent and identically distributed random variables, the sampling distribution of the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

Convergence of random variables

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied.

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