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.     

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$\gamma$.     

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.     

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.     

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).     

A connection between extreme value theory and long time approximation of SDEs

We consider a sequence _(ξn)n≥1${\left({\xi }_n\right)}_{n\ge 1}$ of i.i.d.   random values residing in the domain of attraction of an extreme value distribution. For such a sequence, there exist (_an)$\left(a_n\right)$ and (_bn)$\left(b_n\right)$, with _an>0$a_n>0$ and _bn∈R$b_n\in \mathbb{R}$ for every n≥1$n\ge 1$, such that the sequence (_Xn)$\left(X_n\right)$ defined by _Xn=(max(_ξ1,…,_ξn)−_bn)/_an$X_n=(max({\xi }_1,\dots ,{\xi }_n)-b_n)/a_n$ converges in distribution to a non-degenerated distribution.     

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.     

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.     

Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.     

Nonparametric estimation for pure jump Lévy processes based on high frequency data

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n$n$ discrete time observations with step Δ$\Delta$. The asymptotic framework is: n$n$ tends to infinity, Δ=_Δn$\Delta ={\Delta }_n$ tends to zero while n_Δn$n{\Delta }_n$ tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global L²${\mathbb{L}}^2$-risk. Second, we use a direct approach (“time domain”) which allows us to construct an estimator on a given compact interval. We provide a bound for L²${\mathbb{L}}^2$-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.     

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 ∞$\infty$.