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Soit $T_{\alpha }$ la rotation sur le cercle d’angle irrationnel $\alpha$, soit ${\left(S_k\right)}_{k\ge 0}$ une marche aléatoire transiente sur $\mathbb{Z}$. Soit $f\in L^2\left(\mu \right)$ et $H\in ]0,1[$, nous étudions la convergence faible de la suite $\frac{1}{n^H}{\sum }_{k=0}^{\left[nt\right]-1}f\circ T_{\alpha }^{S_k},n\ge 1.$

Let ${\left(S_n\right)}_{n\ge 0}$ be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let $h$ be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in $D[0,1]$ the set of right continuous real-valued functions with left limits, defined by $$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$ Statistical applications are presented, in particular we prove a strong law of large numbers for $U$-statistics...

Let =( )≥0 be a random walk on ℤ and =( )∈ℤ a stationary random sequence of centered random variables, independent of . We consider a random walk in random scenery that is the sequence of random variables ( )≥0, where =∑=0 , ∈ℕ. Under a weak dependence assumption on the scenery we prove a functional limit theorem generalizing Kesten and Spitzer’s [ (1979) 5–25] theorem.

Let ( be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in the set of right continuous real-valued functions with left limits, defined by $$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$ Statistical applications are presented, in particular we prove a strong law of large numbers for -statistics indexed by a one-dimensional...

We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to dependent random variables sampled by a $\mathbb{Z}$-valued transient random walk. This extends the results obtained by [N. Guillotin-Plantard and D. Schneider, (2003) 477–497]. An application to parametric estimation by random sampling is also provided.

Random walks in random scenery are processes defined by $Z_n:={\sum }_{k=1}^n{\xi }_{X_1+\cdots +X_k}$, where $(X_k,k\ge 1)$ and $({\xi }_y,y\in {\mathbb{Z}}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb{Z}}^d$ and $ℝ$ respectively. We suppose that the distributions of $X_1$ and ${\xi }_0$ belong to the normal basin of attraction of stable distribution of index $\alpha \in (0,2]$ and $\beta \in (0,2]$. When $d=1$ and $\alpha \ne 1$, a functional limit theorem has been established in ( (1979) 5–25) and a local limit theorem in (To appear). In this paper, we establish the convergence in distribution and a local...