Let S=(Sk)k≥0 be a random walk on ℤ and ξ=(ξi)i∈ℤ a stationary random sequence of centered random variables, independent of S. We consider a random walk in random scenery that is the sequence of random variables (Un)n≥0, where Un=∑k=0nξSk, n∈ℕ. Under a weak dependence assumption on the scenery ξ we prove a functional limit theorem generalizing Kesten and Spitzer’s [Z. Wahrsch. Verw. Gebiete50 (1979) 5–25] theorem.

Observations are made on a point process $𝛯$ in ${ℝ}^d$ in a window $Q_{\lambda }$ of volume $\lambda$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,𝛯)$, is a function of the points within a random distance of $x$. When the input $𝛯$ is a Poisson or binomial point process, the large $\lambda$ limit theory for the total score ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input $𝛯$....

Let, for each t∈T, ψ(t, ۔) be a random measure on the Borel σ-algebra in ℝd such that Eψ(t, ℝd)k < ∞ for all k and let $\widehat{\psi }$(t, ۔) be its characteristic function. We call the function $\widehat{\psi }$ (t1,…, tl ; z1,…, zl) = $𝖤{\prod }_{j=1}^l\widehat{\psi }(t_j,z_j)$ of arguments l∈ ℕ, t1, t2… ∈T, z1, z2∈ ℝd the covaristic of the measure-valued random function (MVRF) ψ(۔, ۔). A general limit theorem for MVRF's in terms of covaristics is proved and applied to functions of the kind ψn(t, B) = µ{x : ξn(t, x) ∈B}, where μ is a nonrandom finite measure...

Let, for each t ∈ T, ψ(t, ۔) be a random measure on the Borel σ-algebra in ℝd such that Eψ(t, ℝd)k &lt; ∞ for all kand let $\widehat{\psi }$(t, ۔) be its characteristic function. We call the function $\widehat{\psi }$ (t1,…, tl ; z1,…, zl) = $𝖤{\prod }_{j=1}^l\widehat{\psi }(t_j,z_j)$ of argumentsl ∈ ℕ, t1, t2… ∈ T, z1, z2 ∈ ℝd the covaristic of the measure-valued random function (MVRF) ψ(۔, ۔). A general limit theorem for MVRF's in terms of covaristics is proved and applied to functions of the kind ψn(t, B) = µ{x : ξn(t, x) ∈ B}, where μ is a nonrandom finite measure...

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 (Z. Wahrsch. Verw. Gebiete50 (1979) 5–25) and a local limit theorem in (Ann. Probab.To appear). In this paper, we establish the convergence in...

A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based...

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 indexed by a one-dimensional...

Let (Sn)n≥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 indexed by...

We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.

Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ${\left|\right|f\left|\right|}_{W^{\sigma ,2}}$ of a function f ∈ L²(E,μ) have the property $1/Cℰ(f,f)\le lim in{f}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le lim su{p}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le Cℰ(f,f)$, where ℰ is the Dirichlet form relative to the fractional diffusion.