Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Necessary and sufficient condition for the Lamperti invariance principle

Authors: Alfredas Rackauskas and Charles Suquet
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
Journal: Theor. Probability and Math. Statist. 68 (2004), 127-137
MSC (2000): Primary 60F17; Secondary 60B12
Published electronically: May 24, 2004
MathSciNet review: 2000642
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Abstract: Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of random variables with null expectation and variance one, $S_n:= X_1+\dots + X_n$, and $\xi_n$ the random polygonal line with vertices $(k/n,S_k)$, $k=0,1,\dots,n$. By a theorem of Lamperti (1962), if $X_1$ has a moment of order $p>2$, then $n^{-1/2}\xi_n$ weakly converges to the standard Brownian motion $W$ in the Hölder space $H_\alpha$ for $\alpha<1/2-1/p$. We prove that a necessary and sufficient condition for the $H_\alpha$-weak convergence of this process to $W$ is that $\mathsf{P}(\vert X_1\vert>t)=o(t^{-p(\alpha)})$, where $p(\alpha)=1/(1/2-\alpha)$. As an illustration, we present an application to the change point detection under the epidemic alternative.

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Additional Information

Alfredas Rackauskas
Affiliation: Vilnius University, Department of Mathematics, Naugarduko 24, Lt-2006 Vilnius, Lithuania

Charles Suquet
Affiliation: Université des Sciences et Technologies de Lille, Laboratoire de Mathématiques Appliquées F.R.E. CNRS 2222, Bât. M2, U.F.R. de Mathématiques, F-59655 Villeneuve d’Ascq Cedex France

Received by editor(s): April 4, 2002
Published electronically: May 24, 2004
Additional Notes: Research supported by a cooperation agreement CNRS/LITHUANIA (4714).
Article copyright: © Copyright 2004 American Mathematical Society