Heat-kernel estimates for random walk among random conductances with heavy tail

We study models of discrete-time, symmetric, Z^d${\mathbb{Z}}^d$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances _ωxy∈[0,1]${\omega }_{xy}\in [0,1]$, with polynomial tail near 0 with exponent γ>0$\gamma >0$. We first prove for all d≥5$d\ge 5$ that the return probability shows an anomalous decay (non-Gaussian) that approaches (up to sub-polynomial terms) a random constant times n⁻²$n^{-2}$ when we push the power γ$\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay n^−d/2$n^{-d/2}$ for large values of the parameter γ$\gamma$.     

Moderate deviations for a random walk in random scenery

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d≥2$d\ge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d≥4$d\ge 4$ we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d≥3$d\ge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst for d=2$d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.     

Asymptotic theory for the multidimensional random on-line nearest-neighbour graph

The on-line nearest-neighbour graph on a sequence of n$n$ uniform random points in (0,1)^d${(0,1)}^d$ (d∈N$d\in \mathbb{N}$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent α∈(0,d/2]$\alpha \in (0,d/2]$, we prove O(max{n^1−(2α/d),logn})$O(max\{n^{1-(2\alpha /d)},logn\})$ upper bounds on the variance. On the other hand, we give an n→∞$n\to \infty$ large-sample convergence result for the total power-weighted edge-length when α>d/2$\alpha >d/2$. We prove corresponding results when the underlying point set is a Poisson process of intensity n$n$.     

Monotonicity properties of multi-dimensional reflected diffusions in random environment and applications

We consider an N$N$-dimensional reflected process, modeling an infinite capacity fluid queues network, of which service and input rates depend on the queue levels as well as on the state of an exterior ergodic stationary process. N$N$ is the number of queues in the network. We prove a monotonicity result for such a process, from which we deduce stability results for networks of queues. Particular attention is paid to the case N=2$N=2$. Next, we give some applications of those stability results.     

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α$\alpha$ is in (0,2)$(0,2)$, equal to 2, and in (2,∞)$(2,\infty )$, respectively. The partial sum weakly converges to a functional of α$\alpha$-stable process when α<2$\alpha <2$ and converges to a functional of Brownian motion when α≥2$\alpha \ge 2$. When the process is of short-memory and α<4$\alpha <4$, the autocovariances converge to functionals of α/2$\alpha /2$-stable processes; and if α≥4$\alpha \ge 4$, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α$\alpha$ and β$\beta$ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2$\alpha /2$-stable processes; (ii) Rosenblatt processes (indexed by β$\beta$, 1/2<β<3/4$1/2<\beta <3/4$); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α$\alpha$ and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1]$[0,1]$ with either (i) the _J1$J_1$ or the _M1$M_1$ topology (Skorokhod, 1956); or (ii) the weaker form S$S$ topology (Jakubowski, 1997). Some statistical applications are also discussed.     

Monotonicity of the reflected Bessel transition density on the diagonal

For α∈R$\alpha \in \mathbb{R}$, let p^R(t,x,x)$p^R(t,x,x)$ denote the diagonal of the transition density of the α$\alpha$-Bessel process in (0,1]$(0,1]$, killed at 0 and reflected at 1. As a function of x$x$, if either α≥3$\alpha \ge 3$ or α=1$\alpha =1$, then for t>0$t>0$, the diagonal is nondecreasing. This monotonicity property fails if 1≠α<3$1\ne \alpha <3$.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

Approximate martingale estimating functions for stochastic differential equations with small noises

An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter ε$\epsilon$ from discrete time observations at n$n$ regularly spaced time points k/n$k/n$, k=0,1,…,n$k=0,1,\dots ,n$. We show asymptotic efficiency of an M$M$-estimator derived from the approximate martingale estimating function as ε→0$\epsilon \to 0$ and n→∞$n\to \infty$ simultaneously.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn^−β$\alpha n^{-\beta }$, with α>0$\alpha >0$, β∈[0,+∞]$\beta \in [0,+\infty ]$ and n$n$ is the scaling parameter. Depending on the regime of β$\beta$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1$\beta =1$, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α$\alpha$, interpolating a fractional Brownian motion of Hurst exponent 1/4$1/4$ and the degenerate process equal to zero.     

On the equivalence of the static and dynamic points of view for diffusions in a random environment

We study the equivalence of the static and dynamic points of view for diffusions in a random environment in dimension one. First we prove that the static and dynamic distributions are equivalent if and only if either the speed in the law of large numbers does not vanish, or b/a$b/a$ is a.s. the gradient of a stationary function, where a$a$ and b$b$ are the covariance coefficient resp. the local drift attached to the diffusion. We moreover show that the equivalence of the static and dynamic points of view is characterized by the existence of so-called “almost linear coordinates”.     

Conditional residual lifetimes of coherent systems

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n$n$ independent and identically distributed (i.i.d.) components under the condition that at least j$j$ and at most k−1$k-1$ (j<k$j<k$) components have failed by time t$t$. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.     

Limit laws for UGROW random graphs

Representations are found for a limit law L(Z(k,p))$L\left(Z(k,p)\right)$ obtained from an expanding sequence of random forests containing n$n$ nodes with p∈(0,1]$p\in (0,1]$ a probability controlling bond formation. One implies that Z(k,p)$Z(k,p)$ is stochastically decreasing as k$k$ increases and that norming gives an exponential limit law. Limit theorems are given for the order of component trees. The proofs exploit properties of the gamma function.     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).