We continue here the study of free extreme values begun in Ben Arous and Voiculescu (2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu (1998), Barndorff-Nielsen and Thorbjornsen (2005)). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (2006) and give the limit laws for free order statistics.
With motivation from K. Dȩbicki and P. Kisowski (2007), in this paper we derive the exact tail asymptotics of $α(t)$-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance functions lead to qualitatively new results.
In this paper, we study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.
We study translation-invariant determinantal random point fields on the real line. We prove, under quite general conditions, that the smallest nearest spacings between the particles in a large interval have Poisson statistics as the length of the interval goes to infinity.
Based on previous work of Paul Ressel and myself, I show that the space of all "continuous" exchangeable probability measures on a certain set of order processes is a Bauer simplex.
In this paper, we continue Voiculescu's recent work on the analogous extreme value theory in the context of bi-free probability theory. We derive various equivalent conditions for a bivariate distribution function to be bi-freely max-infinitely divisible. A bi-freely max-infinitely divisible distribution function can be expressed in terms of its marginals and a special form of copulas. Such a distribution function is shown to be also max-infinitely divisible in the classical sense. In addition, we characterize the set of bi-free extreme value distribution functions. A distribution function of …
We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable L{é}vy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.
Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0; 1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian's lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.
Path decomposition is performed to characterize the law of the pre/post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T: As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre/postsupremum, post-infimum processes and the intermediate processes are obtained together with the law of drawdown durations.
Free probability analogs of the basics of extreme-value theory are obtained, based on Ando's spectral order. This includes classification of freely max-stable laws and their domains of attraction, using ``free extremal convolutions'' on the distributions. These laws coincide with the limit laws in the classical peaks-over-threshold approach. A free extremal projection-valued process over a measure-space is constructed, which is related to the free Poisson point process.
The seminal papers of Pickands [1,2] paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting obtaining the exact asymptotics of the exceedance probabilities for both stationary…
This paper gives a new representation of Pickands' constants, which arise in the study of extremes for a variety of Gaussian processes. Using this representation, we resolve the long-standing problem of devising a reliable algorithm for estimating these constants. A detailed error analysis illustrates the strength of our approach.
We investigate a family of multiple-stable processes that may exhibit either long-range or short-range dependence, depending on the parameters. There are two parameters for the processes, the memory parameter $β\in(0,1)$ and the multiplicity parameter $p\in\mathbb N$. We investigate the macroscopic limit of extremes of the process, in terms of convergence of random sup-measures, for the full range of parameters. Our results show that (i) the extremes of the process exhibit long-range dependence when $β_p := pβ-p+1\in(0,1)$, with a new family of random sup-measures arising in the limit, (ii) th…
Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t)), t\in [0,S]$ be a Gaussian vector process and let $g(\boldsymbol{x}),\boldsymbol{x}\in\mathbb{R}^d$ be a continuous homogeneous function. In this paper we are concerned with the exact tail asymptotics of the chaos process $g(\boldsymbol{X}(t))+ h(t),t\in [0,S]$ with trend function $h$. Both scenarios $\boldsymbol{X}(t)$ is locally-stationary and $\boldsymbol{X}(t)$ is non-stationary are considered. Important examples include the product of Gaussian processes and chi-processes.
For $X_i(t), i=1,\ldots, n, t\in [0,T]$ centered Gaussian processes, the chi-square process $\sum_{i=1}^{n}X_i^2(t)$ appears naturally as limiting processes in various statistical models. In this paper, we are concerned with the exact tail asymptotics of the supremum taken over discrete grids of a class of locally stationary chi-square processes where $X_i(t),\ 1\leq i\leq n$ are not identical. An important tool for establishing our results is a generalisation of Pickands lemma under the discrete scenario. An application related to the change-point problem is discussed.
We obtain limit theorems for the row extrema of a triangular array of zero-modified geometric random variables. Some of this is used to obtain limit theorems for the maximum family size within a generation of a simple branching process with varying geometric offspring laws.
We give conditions under which the tail probability of the supremum over unit interval of a Levy process with light tail is equivalent to the tail of the value of the process at the right endpoint.
For every $n\in\N$, let $X_{1n},..., X_{nn}$ be independent copies of a zero-mean Gaussian process $X_n=\{X_n(t), t\in T\}$. We describe all processes which can be obtained as limits, as $n\to\infty$, of the process $a_n(M_n-b_n)$, where $M_n(t)=\max_{i=1,...,n} X_{in}(t)$ and $a_n, b_n$ are normalizing constants. We also provide an analogous characterization for the limits of the process $a_nL_n$, where $L_n(t)=\min_{i=1,...,n} |X_{in}(t)|$.
The time it takes the fastest searcher out of $N\gg1$ searchers to find a target determines the timescale of many physical, chemical, and biological processes. This time is called an extreme first passage time (FPT) and is typically much faster than the FPT of a single searcher. Extreme FPTs of diffusion have been studied for decades, but little is known for other types of stochastic processes. In this paper, we study the distribution of extreme FPTs of piecewise deterministic Markov processes (PDMPs). PDMPs are a broad class of stochastic processes that evolve deterministically between random…
We use point processes theory to describe the asymptotic distribution of all upper order statistics for observations collected at renewal times. As a corollary, we obtain limiting theorems for corresponding extremal processes.
We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the Poisson process. We use a coupling of the point process with its Palm version to prove a general Poisson limit theorem. We then combine our general result with the theory of asymptotic shapes of large cells (Kendall's problem) in random mosaics and prove Poisson limit theorems for large cells (with respect to a general size functional) in the Poisson-Vorono…
This paper studies the supremum of a chi-square process with trend over a threshold-dependent-time horizon. Under the assumption that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square process with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.
The article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster and then we apply the developed theory to determine the limiting distribution of $\max\{X_j: j=0,\ldots, K(t)\}$, where $K(t)$ is the number of i.i.d. observations $(X_j)$ arriving up to the time $t$ according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.
We investigate relations between additive convolution semigroup and max-convolution semigroup through the law of large numbers for the free multiplicative convolution. Based on the relation, we give a formula related with Belinschi-Nica semigroup and max-Belinschi-Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.
The $q$-Ornstein-Uhlenbeck processes, $q\in(-1,1)$, are a family of stationary Markov processes that converge weakly to the standard Ornstein-Uhlenbeck process as $q$ tends to 1. It has been noticed recently that in terms of path properties, however, for each $q$ fixed the $q$-Ornstein-Uhlenbeck process behaves qualitatively different from their Gaussian counterpart in several aspects. Here, two limit theorems on the extremes of $q$-Ornstein-Uhlenbeck processes are established. Both results are based on the weak convergence of the tangent process at the lower boundary, a positive self-similar …
We develop and generalize the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to {\em sequential dynamical systems}, given by uniformly expanding maps, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail.
We study the partial maxima of stationary α-stable processes. We relate their asymptotic behavior to the ergodic theoretical properties of the flow. We observe a sharp change in the asymptotic behavior of the sequence of partial maxima as flow changes from being dissipative to being conservative, and argue that this may indicate a change from a short memory process to a long memory process.
The extremal behaviour of a Markov chain is typically characterized by its tail chain. For asymptotically dependent Markov chains existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region and for asymptotically independent chains recent results fail to cover well-known asymptotically independent processes such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including …
This paper considers extreme values attained by a centered, multidimensional Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift $d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity conditions, we establish the asymptotics of \[\log\mathbb P\left(\exists{t\in T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right),\] for positive thresholds $q_i>0$, $i=1,\ldots,n$, and $u\to\infty$. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.
We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.
Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg …
We compute the bi-free max-convolution which is the operation on bi-variate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bi-partite two-faced pairs of hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely-divisible laws their determination becomes in this way a classical analysis question.
We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the $\limsup$ and a law of the triple logarithm for the $\liminf$. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several queuing systems and a birth and death process.
In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. Unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution …
This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Br…
For each $n\geq 1$, let $ {X_{in}, \quad i \geq 1} $ be independent copies of a nonnegative continuous stochastic process $X_{n}=(X_n(t))_{t\in T}$ indexed by a compact metric space $T$. We are interested in the process of partial maxima [\tilde M_n(u,t) =\max \{X_{in}(t), 1 \leq i\leq [nu]},\quad u\geq 0,\ t\in T.] where the brackets $[\,\cdot\,]$ denote the integer part. Under a regular variation condition on the sequence of processes $X_n$, we prove that the partial maxima process $\tilde M_n$ weakly converges to a superextremal process $\tilde M$ as $n\to\infty$. We use a point process app…
Max-stable processes are widely used to model spatial extremes. These processes exhibit asymptotic dependence meaning that the large values of the process can occur simultaneously over space. Recently, inverted max-stable processes have been proposed as an important new class for spatial extremes which are in the domain of attraction of a spatially independent max-stable process but instead they cover the broad class of asymptotic independence. To study the extreme values of such processes we use the conditioned approach to multivariate extremes that characterises the limiting distribution of …
As a useful and elegant tool of extreme value theory, the study of point processes on a metric space is important and necessary for the analyses of heavy-tailed functional data. This paper focuses on the definition and properties of such point processes. A complete convergence result for a regularly varying iid sequence in a metric space is proved as an example of the application in extreme value theory.
Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t))$ be a Gaussian vector process and $g(t)$ be a continuous function. The asymptotics of distribution of $\left\|\boldsymbol{X}(t)\right\|_p$, the $L^p$ norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of $\left\|\boldsymbol{X}(t)\right\|^c_p,\ c>0, $ with trend $g(t)$ over $[0,T]$. Both scenarios that $\boldsymbol{X}(t)$ is locally stationary and non-stationary are considered. Important examples include $\sum_{i=1}^d \left|X_i(t)\right|+g…
This paper provides the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the…
In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic deformation, water flow, electric-magnetic fields, temperature distribution, etc. A random coefficient is introduced to account for the system's uncertainty and/or imperfect measurements. This random coefficient is described by a Gaussian process (the input process) and thus the solution to the differential equation (under certain boundary conditions) is a complex…
Let $X(t)=(X_1(t), \dots, X_n(t)), t\in \mathcal{T}\subset \mathbb{R} $ be a centered vector-valued Gaussian process with independent components and continuous trajectories, and $h(t)=(h_1(t),\dots, h_n(t)), t\in \mathcal{T} $ be a vector-valued continuous function. We investigate the asymptotics of $$\mathbb{P}\left(\sup_{t\in \mathcal{T} } \min_{1\leq i\leq n}(X_i(t)+h_i(t))>u\right)$$ as $u\to\infty$. As an illustration to the derived results we analyze two important classes of $X(t)$: with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymp…
We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed points and prove the existence of Extreme Value Laws for such processes. We use an approach developed in \cite{FFV16}, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps obtained in \cite{FFV16}.
We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the co…
Let $\{ζ_{m,k}^{(κ)}(t), t \ge0\}, κ>0$ be random processes defined as the differences of two independent stationary chi-type processes with $m$ and $k$ degrees of freedom. In applications such as physical sciences and engineering dealing with structure reliability, of interest is the approximation of the probability that the random process $ζ_{m,k}^{(κ)}$ stays in some safety region up to a fixed time $T$. In this paper we derive the asymptotics of $\mathbb{P}\{\sup_{t\in[0, T]}ζ_{m,k}^{(κ)}(t)> u\}, {u\to\infty}$ under some assumptions on the covariance structures of the underlying Gaussian …
Using the proposed by us thinning approach to describe extreme matrices, we find an explicit exponentiation formula linking classical extreme laws of Fréchet, Gumbel and Weibull given by Fisher-Tippet-Gnedenko classification and free extreme laws of free Fréchet, free Gumbel and free Weibull by Ben Arous and Voiculescu [1]. We also develop an extreme random matrix formalism, in which refined questions about extreme matrices can be answered. In particular, we demonstrate explicit calculations for several more or less known random matrix ensembles, providing examples of all three free extreme la…
Noncolliding diffusion processes reported in the present paper are $N$-particle systems of diffusion processes in one-dimension, which are conditioned so that all particles start from the origin and never collide with each other in a finite time interval $(0, T)$, $0 < T < \infty$. We consider four temporally inhomogeneous processes with duration $T$, the noncolliding Brownian bridge, the noncolliding Brownian motion, the noncolliding three-dimensional Bessel bridge, and the noncolliding Brownian meander. Their particle distributions at each time $t \in [0, T]$ are related to the eigenvalue di…
We consider two independent random variables with the given tail asymptotic (e.g. power or exponential). We find tail asymptotic for their sum and product. This is done by some cumbersome but purely technical computations and requires the use of the Laplace method for asymptotic of integrals. We also recall the results for asymptotic of a self-similar locally stationary centered Gaussian process plus a deterministic drift; and we find the asymptotic for the same probability after multiplying the drift by a random variable, which is independent of this process. Keywords: tail asymptotic, Laplac…
In this contribution we are concerned with the asymptotic behaviour as $u\to \infty$ of $\mathbb{P}\{\sup_{t\in [0,T]} X_u(t)> u\}$, where $X_u(t),t\in [0,T],u>0$ is a family of centered Gaussian processes with continuous trajectories. A key application of our findings concerns $\mathbb{P}\{\sup_{t\in [0,T]} (X(t)+ g(t))> u\}$ as $u\to\infty$, for $X$ a centered Gaussian process and $g$ some measurable trend function. Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.
For $\{X(t), t \in G_δ\}$ a centered Gaussian process with stationary increments and a.s. sample paths on a discrete grid $G_δ=\{0,δ,2δ, ...\}$, where $δ>0$, we investigate the stationary reflected process $$Q_{δ,X}(t) = \sup\limits_{s\in [t,\infty)\cap G_δ}\big( X(s)-X(t)-c(s-t)\big), \ t \in G_δ$$ with $c>0$. We derive the exact asymptotics of $$\mathbb P\{\sup\limits_{t\in [0,T]\cap G_δ} Q_{δ,X}(t)>u\} \quad \text{and} \quad \mathbb P\{\inf\limits_{t\in [0,T]\cap G_δ} Q_{δ,X}(t)>u\},$$ as $u\to\infty$, with $T>0$. It appears that $φ=\lim_{u\to\infty} \frac{σ^2(u)}{u}$ determines the asympto…
Extreme events are an important theme in various areas of science because of their typically devastating effects on society and their scientific complexities. The latter is particularly true if the underlying dynamics does not lead to independent extreme events as often observed in natural systems. Here, we focus on this case and consider stationary stochastic processes that are characterized by long-range memory and heavy-tailed distributions, often called fractional Lévy noise. While the size distribution of extreme events is not affected by the long-range memory in the asymptotic limit and …
We use the Stein-Chen method to study the extremal behaviour of the problem of extremes for univariate and bivariate geometric laws. We obtain a rate for the convergence to the Gumbel distribution of the law of the maximum of i. i. d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marsh…
We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised $k$th order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random va…
We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and an estimate for the right tail up to a multiplicative constant on the law of the centered maximum.
