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


[1] M. Donsker, "An invariance principle for certain probability limit theorems" Memoirs Amer. Math. Soc. , 6 (1951) pp. 1–12
[2] Yu.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Theor. Probab. Appl. , 1 (1956) pp. 157–214 Teor. Veroyatnost. Prilozhen. , 1 : 2 (1956) pp. 177–238
[3] P. Billingsley, "Convergence of probability measures" , Wiley (1968)



[a1] L.P. Breiman, "Probability" , Addison-Wesley (1968)
How to Cite This Entry:
Invariance, principle of. Encyclopedia of Mathematics. URL:,_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