View source for Invariance, principle of

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521901.png" /> be independent identically-distributed real-valued random variables with zero expectation and variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521902.png" />; consider the random polygonal line

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521903.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521904.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521905.png" /> is a real-valued continuous function on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521906.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521907.png" /> with the supremum norm (or only continuous everywhere except on a set of Wiener measure zero), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521908.png" /> converges in distribution to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i0521909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219010.png" /> is a Wiener random function. Thus, the limiting distribution for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219011.png" /> does not depend on any special properties of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219012.png" />.

A typical scheme for the use of the invariance principle consists in finding the limiting distribution for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219013.png" /> by finding the limiting distribution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219015.png" /> is a random polygonal line constructed in the same way as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219016.png" /> from some specially chosen sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219017.png" />. For example, if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219018.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219019.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219020.png" />, and, since

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219021.png" /></td> </tr></table>

one has that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219022.png" /></td> </tr></table>

converges in distribution to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219023.png" />. To find the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219024.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219025.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219026.png" /> is used, and as a result of the calculations one obtains

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052190/i05219027.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Donsker,   "An invariance principle for certain probability limit theorems"  ''Memoirs Amer. Math. Soc.'' , '''6'''  (1951)  pp. 1–12</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Billingsley,   "Convergence of probability measures" , Wiley  (1968)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.P. Breiman,   "Probability" , Addison-Wesley  (1968)</TD></TR></table>