The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

Reflection principle and Ocone martingales

Let M=_(Mt)t≥0$M={\left(M_t\right)}_{t\ge 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence _(an)n≥1${\left(a_n\right)}_{n\ge 1}$ of real numbers which converges to 0 and such that M$M$ satisfies the reflection property at all levels _an$a_n$ and 2_an$2a_n$ with n≥1$n\ge 1$, then M$M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels _an$a_n$? We prove that this question is equivalent to the fact that for Brownian motion, the σ$\sigma$-field of the invariant events by all reflections at levels _an$a_n$, n≥1$n\ge 1$ is trivial. We establish similar results for skip free Z$\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretization in space.     

Randomly weighted self-normalized Lévy processes

Let (_Ut,_Vt)$(U_t,V_t)$ be a bivariate Lévy process, where _Vt$V_t$ is a subordinator and _Ut$U_t$ is a Lévy process formed by randomly weighting each jump of _Vt$V_t$ by an independent random variable _Xt$X_t$ having cdf F$F$. We investigate the asymptotic distribution of the self-normalized Lévy process _Ut/_Vt$U_t/V_t$ at 0 and at ∞$\infty$. We show that all subsequential limits of this ratio at 0 (∞$\infty$) are continuous for any nondegenerate F$F$ with finite expectation if and only if _Vt$V_t$ belongs to the centered Feller class at 0 (∞$\infty$). We also characterize when _Ut/_Vt$U_t/V_t$ has a non-degenerate limit distribution at 0 and ∞$\infty$.     

Median,concentration and fluctuations for Lévy processes

We estimate a median of f(_Xt)$f\left(X_t\right)$ where f$f$ is a Lipschitz function, X$X$ is a Lévy process and t$t$ is an arbitrary time. This leads to concentration inequalities for f(_Xt)$f\left(X_t\right)$. In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

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.     

Asymptotics of a Brownian ratchet for protein translocation

Protein translocation in cells has been modelled by Brownian ratchets  . In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ with a changing reflection point _(Rt)t≥0${\left(R_t\right)}_{t\ge 0}$. The rate of change of _Rt$R_t$ is γ(_Xt−_Rt)$\gamma (X_t-R_t)$ and the new reflection boundary is distributed uniformly between _Rt−$R_{t-}$ and _Xt$X_t$. The asymptotic speed of the ratchet scales with γ^1/3${\gamma }^{1/3}$ and the asymptotic variance is independent of γ$\gamma$.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

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.     

Transformation formulas for fractional Brownian motion

We derive a Molchan–Golosov-type integral transform which changes fractional Brownian motion of arbitrary Hurst index K$K$ into fractional Brownian motion of index H$H$. Integration is carried out over [0,t]$[0,t]$, t>0$t>0$. The formula is derived in the time domain. Based on this transform, we construct a prelimit which converges in L²(P)$L^2\left(\mathbb{P}\right)$-sense to an analogous, already known Mandelbrot–Van Ness-type integral transform, where integration is over (−∞,t]$(-\infty ,t]$, t>0$t>0$.     

Density estimation for compound Poisson processes from discrete data

In this article we investigate the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over [0,T]$[0,T]$. We consider the case where the sampling rate Δ=_ΔT→0$\Delta ={\Delta }_T\to 0$ as T→∞$T\to \infty$. We propose an adaptive wavelet threshold density estimator and study its performance for _Lp$L_p$ losses, p≥1$p\ge 1$, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates _ΔT${\Delta }_T$ that vanish slowly. The estimation procedure is based on the explicit inversion of the operator giving the law of the increments as a nonlinear transformation of the jump density.     

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$\gamma$.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at x$x$ given by _μ1(x)→0${\mu }_1\left(x\right)\to 0$ as x→∞$x\to \infty$ is known as Lamperti’s problem  . We give sharp almost-sure bounds for processes of this type in the case where _μ1(x)${\mu }_1\left(x\right)$ is of order x^−β$x^{-\beta }$ for some β∈(0,1)$\beta \in (0,1)$. The bounds are of order t^1/(1+β)$t^{1/(1+\beta )}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)$(2+2\beta +\epsilon )$-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x^β_μ1(x)$x^{\beta }{\mu }_1\left(x\right)$ has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0$\beta =0$. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.     

Generalized Hermite processes,discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t),t≥0}$\{Z\left(t\right),t\ge 0\}$ called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1)$H\in (1/2,1)$, and include Hermite processes as a special case. They are defined through a homogeneous kernel g$g$, called the “generalized Hermite kernel”, which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels g$g$ can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X(n)}$\left\{X\left(n\right)\right\}$. In addition, we consider a fractionally-filtered version Z^β(t)$Z^{\beta }\left(t\right)$ of Z(t)$Z\left(t\right)$, which allows H∈(0,1/2)$H\in (0,1/2)$. Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.