Asymptotic distribution

A sequence of distributions corresponds to a sequence of random variables Z_i for i = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of Z_i 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 Z_i = 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 Z_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} then Z_i 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

${\displaystyle P\left({\frac {Z_{n}-a_{n}}{b_{n}}}\leq x\right)\approx F(x),}$

${\displaystyle P(Z_{n}\leq z)\approx F\left({\frac {z-a_{n}}{b_{n}}}\right).}$

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 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 ${\displaystyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty }$. 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 ${\displaystyle N(0,\sigma ^{2})}$:[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.

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