Asymptotic distribution

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 ${\displaystyle \{X_1,X_2,\dots \}}$ is a sequence of i.i.d. random variables with ${\displaystyle \mathrm{E}[X_i]=\mu }$ and ${\displaystyle \mathrm{Var}[X_i]={\sigma }^2<\mathrm{\infty }}$. 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 a normal ${\displaystyle N(0,{\sigma }^2)}$:

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

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

Barndorff-Nielson & Cox provide a direct definition of asymptotic normality:

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 distribution. The constants ${\displaystyle a_n}$, ${\displaystyle b_n}$ are called respectively the asymptotic mean and standard deviation of ${\displaystyle V_n}$.