Inmathematics andstatistics, anasymptotic distribution is aprobability 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 thecumulative distribution functions of statisticalestimators.

A sequence of distributions corresponds to asequence ofrandom variablesZ_i for${\displaystyle i=1,2,\dots }$. In the simplest case, an asymptotic distribution exists if the probability distribution ofZ_i converges to a probability distribution (the asymptotic distribution) asi increases: seeconvergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables is always zero orZ_i = 0 asi approaches infinity. Here the asymptotic distribution is adegenerate distribution, corresponding to the value zero.

However, the most usual sense in which the term asymptotic distribution is used arises where the random variablesZ_i are modified by two sequences of non-random values. Thus if

${\displaystyle Y_i=\frac{Z_i-a_i}{b_i}}$

converges in distribution to a non-degenerate distribution for two sequences {a_i} and {b_i} thenZ_i is said to have that distribution as its asymptotic distribution. If the distribution function of the asymptotic distribution isF then, for largen, the following approximations hold

${\displaystyle P\left(\frac{Z_n-a_n}{b_n}\le x\right)\approx F(x),}$

${\displaystyle P(Z_n\le z)\approx F\left(\frac{z-a_n}{b_n}\right).}$

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.

Perhaps the most common distribution to arise as an asymptotic distribution is thenormal distribution. In particular, thecentral limit theorem provides an example where the asymptotic distribution is thenormal distribution.

Central limit theorem

Suppose${\displaystyle \{X_1,X_2,\dots \}}$ is a sequence ofi.i.d. random variables with${\displaystyle \mathrm{E}[X_i]=\mu }$ and. Let${\displaystyle S_n}$ be the average of${\displaystyle \{X_1,\dots ,X_n\}}$. Then as${\displaystyle n}$ approaches infinity, the random variables${\displaystyle \sqrt{n}(S_n-\mu )}$converge in distribution to anormal${\displaystyle N(0,{\sigma }^2)}$:^[1]

The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.

Local asymptotic normality is a generalization of the central limit theorem. It is a property of a sequence ofstatistical models, which allows this sequence to be asymptotically approximated by anormal location model, after a rescaling of the parameter. An important example when the local asymptotic normality holds is in the case ofindependent and identically distributed sampling from aregular parametric model; this is just the central limit theorem.

Barndorff-Nielson & Cox provide a direct definition of asymptotic normality:^[2]

The sequence${\displaystyle \{V_n\}}$ is asymptotically normal if there exist sequences of constants${\displaystyle \{a_n\}}$,${\displaystyle \{b_n\}}$ such that${\displaystyle (V_n-a_n)/b_n}$ converges in distribution to the standard normal distribuion. The constants${\displaystyle a_n}$,${\displaystyle b_n}$ are called respectively the asymptotic mean and standard deviation of${\displaystyle V_n}$.

• Asymptotic analysis
• Asymptotic theory (statistics)
• de Moivre–Laplace theorem
• Limiting density of discrete points
• Delta method

• Types of probability distributions
• Asymptotic theory (statistics)

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• Short description matches Wikidata