Asymptotic Distribution - Definition

Definition

A sequence of distributions corresponds to a sequence of random variables Zi for i = 1, 2, ... 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 always approaches zero—that is, the Zi go to 0 as i goes to 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.

Read more about this topic:  Asymptotic Distribution

Famous quotes containing the word definition:

    “One definition of man is “an intelligence served by organs.””
    —Ralph Waldo Emerson (1803–1882)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)