Asymptotic distribution

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In mathematics and statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables

Z_i

for i = 1 to n for some positive integer n. An asymptotic distribution allows i to range without bound, that is, n is infinite.

A special case of an asymptotic distribution is when the late entries go to zero -- that is, the Z_i go to 0 as i goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.

This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.

An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation

y = 1/x,

y becomes arbitrarily small in magnitude as x increases.

It is often used in time series analysis.