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Limit theorems for U-statistics indexed by a onedimensional random walk
Published online by Cambridge University Press: 15 November 2005
Abstract
Let (Sn)n≥0 be a $\mathbb Z$ 
-random walk and $(\xi_{x})_{x\in \mathbb Z}$ 
be a sequence of independent andidentically distributed $\mathbb R$ 
-valued random variables,independent of the random walk. Let h be a measurable, symmetricfunction defined on $\mathbb R^2$ 
with values in $\mathbb R$ . We study theweak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$ 
, withvalues in D[0,1] the set of right continuous real-valuedfunctions with left limits, defined by \[ \sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1].\] 
Statistical applications are presented, in particular we prove a strong law of large numbersfor U-statistics indexed by a one-dimensional random walk using a result of [1].
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- © EDP Sciences, SMAI, 2005
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