# Invariance, principle of

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Let be independent identically-distributed real-valued random variables with zero expectation and variance ; consider the random polygonal line where . If is a real-valued continuous function on the space of continuous functions on with the supremum norm (or only continuous everywhere except on a set of Wiener measure zero), then converges in distribution to , where is a Wiener random function. Thus, the limiting distribution for the does not depend on any special properties of the .

A typical scheme for the use of the invariance principle consists in finding the limiting distribution for the by finding the limiting distribution for , where is a random polygonal line constructed in the same way as from some specially chosen sequence . For example, if then is continuous on , and, since one has that converges in distribution to . To find the distribution of , the sequence  is used, and as a result of the calculations one obtains How to Cite This Entry:
Invariance, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariance,_principle_of&oldid=13150
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article