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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 and identically distributed $\mathbb R$-valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on $\mathbb R^2$ with values in $\mathbb R$. We study the weak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$, with values in D[0,1] the set of right continuous real-valued functions 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 numbers for 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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