We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64, 193--208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of càdlàg functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange segments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.
Abel-Tauberian theorems relate power law behavior of distributions and their transforms. We formulate and prove a multivariate version for non-standard regularly varying measures on $\mathbb{R}_+^p$ and then apply it to prove that the joint distribution of in- and out-degree in a directed edge preferential attachement model has jointly regularly varying tails.
A spectral representation for regularly varying Lévy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail giving also new insight in the $L^2$-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying Lévy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.
A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the "time change …
Let F be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F. An application to ruin probabilities is developed.
In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the $n$-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the $n$-th generation.
Let $X$ be a Lévy process with regularly varying Lévy measure $ν$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but…
We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main resu…
We introduce the cluster index of a multivariate regularly varying stationary sequence and characterize the index in terms of the spectral tail process. This index plays a major role in limit theory for partial sums of regularly varying sequences. We illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary Markov chain under a drift condition.
A multivariate, stationary time series is said to be jointly regularly varying if all its finite-dimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution of the rescaled series given that, at a fixed time instant, its distance to the origin exceeds a threshold tending to infinity. The limit object, called the tail process, admits a decomposition in independent radial and angular components. Under an appropriate mixing condition, this tail process allows for a concise and explicit description of the…
We consider the functional regular variation in the space $\mathbb{D}$ of càdlàg functions of multivariate mixed moving average (MMA) processes of the type $X_t = \int\int f(A, t - s) Λ(d A, d s)$. We give sufficient conditions for an MMA process $(X_t)$ to have càdlàg sample paths. As our main result, we prove that $(X_t)$ is regularly varying in $\mathbb{D}$ if the driving Lévy basis is regularly varying and the kernel function $f$ satisfies certain natural (continuity) conditions. Finally, the special case of supOU processes, which are used, e.g., in applications in finance, is considered i…
We propose a class of strongly efficient rare event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on the heavy-tailed sample path large deviations result recently established in Rhee, Blanchet, and Zwart (2016). The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are "universal" in the…
For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to …
We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index $α$ of the right tail lies in the interval $(1, \, 3/2)$.
We study a linear recursion with random Markov-dependent coefficients. In a "regular variation in, regular variation out" setup we show that its stationary solution has a multivariate regularly varying distribution. This extends results previously established for i.i.d. coefficients.
A notion of tail dependence based on operator regular variation is introduced for copulas, and the standard tail dependence used in the copula literature is included as a special case. The non-standard tail dependence with marginal power scaling functions having possibly distinct tail indexes is investigated in detail. We show that the copulas with operator tail dependence, incorporated with regularly varying univariate margins, give rise to a rich class of the non-standard multivariate regularly varying distributions. We also show that under some mild conditions, the copula of a non-standard …
We study the extremes of multivariate regularly varying random fields. The crucial tools in our study are the tail field and the spectral field, notions that extend the tail and spectral processes of Basrak and Segers (2009). The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown-Resnick random fields.
We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in $[0,\infty)^d$ under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure.
We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the $n$-th generation when the tail of the displacement behaves like $\exp(-K(x))$, where either $K$ is a regularly varying function of index $r> 0$, or $K$ …
It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently, \cite{W} indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the $d$-dimensional lattice torus (with size $n^d$) according to \cite{W}, and show that the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in $\ln n$.
In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the $n$-th generation) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Fre…
We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regularly varying with index $α>0$. For predictable integrands with a finite $(α+δ)$-moment, for some $δ>0$, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy process and we determine its limit measure associated with regular variation on the space of càdlàg functions.
Let $\{\mm_n, n=0,1,...\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\mm_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, $\su |W-W_n|$, $\su |W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that $\rP\{W_1>x\}$ regularly varies at $\infty$ it is proved that $\rP\{AW_n>x\}$ regularly varies at $\infty$ with the same exponent. All the proofs given …
We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.
This article studies asymptotic approximations of ruin probabilities of multivariate random walks with heavy-tailed increments. Under our assumptions, the distributions of the increments are closely connected to multivariate subexponentiality and admit dependence between components. Keywords: subexponential distribution, ruin probability, multivariate random walk
We show a new functional limit theorem for weakly dependent regularly varying sequences of random vectors. As it turns out, the convergence takes place in the space of R^d valued càdlàg functions endowed with the so-called weak M1 topology. The theory is illustrated on two examples. In particular, we demonstrate why such an extension of Skorohod's M1 topology is actually necessary for the limit theorem to hold.
We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and …
In this article subordination of random walks in $R^d$ is considered. We prove that subordination of random walks in the sense of [BSC12] yields the same process as subordination of Lévy processes (in the sense of Bochner). Furthermore, we prove that appropriately scaled subordinate random walk converges to a multiple of a rotationally $2α$-stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero with index $α\in (0,1]$.
This paper extends the result of Broniatowski and Caron (2013) pertaining to the asymptotic distribution of a random walk conditioned on its final value as the number of summands increase. We consider multivariate light-tailed random walk and present a sharp approximation of long runs conditioned by an average of a function of its summands as their number tends to infinity.
A random vector $X$ with representation $X=\sum_{j\geq0}A_jZ_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, `predictable' with respect to the sequence $(Z_j)$. The distribution of $Z_1$ is assumed to be multivariate regular varying. Moment conditions on the matrices $(A_j)$ are determined under which the distribution of $X$ is regularly varying and, in fact, `inherits' its regular variation from that of the $(Z_j)$'s. We compute the associated limiting measure. Examples include linear process…
Cluster indices describe extremal behaviour of stationary time series. We consider runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain central limit theorems under conditions that can be easily verified for a large class of models. In particular, we show that blocks and runs estimators have the same limiting variance.
We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.
Let $Φ_n$ be an i.i.d. sequence of Lipschitz mappings of $\R^d$. We study the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the recursion $X_n^x = Φ_n(X^x_{n-1})$, $n\in\N$, $X_0^x=x\in\R^d$. We assume that $Φ_n(x)=Φ(A_n x,B_n(x))$ for a fixed continuous function $Φ:\R^d\times \R^d\to\R^d$, commuting with dilations and i.i.d random pairs $(A_n,B_n)$, where $A_n\in {End}(\R^d)$ and $B_n$ is a continuous mapping of $\R^d$. Moreover, $B_n$ is $α$-regularly varying and $A_n$ has a faster decay at infinity than $B_n$. We prove that the stationary measure $ν$ of the Markov chain $\{X_n^…
Multivariate stochastic recurrence equations (SREs) are investigated when coefficients are triangular matrices. If coefficient matrices of SREs have all strictly positive elements, the Kesten's classical result yields solutions with regularly varying tails such that the tail indices of solutions are the same through coordinates. This framework is too restrictive for applications. In order to widen the applicability of the SREs, we study SREs with triangular matrix coefficients and prove that they have regularly varying solutions which may exhibit coordinate-wisely different tail exponents. We …
This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $α<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a c…
Multivariate max-stable processes are important for both theoretical investigations and various statistical applications motivated by the fact that these are limiting processes, for instance of stationary multivariate regularly varying time series, [1]. In this contribution we explore the relation between homogeneous functionals and multivariate max-stable processes and discuss the connections between multivariate max-stable process and zonoid / max-zonoid equivalence. We illustrate our results considering Brown-Resnick and Smith processes.
We study a random walk $\mathbf{S}_n$ on $\mathbb{Z}^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbolα=(α_1,\ldots,α_d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\mathbf{0},\mathbf{x})$ as $\|\mathbf{x}\|\to +\infty$, along the "favorite direction or scaling": (i) if $\sum_{i=1}^d α_i^{-1} < 2$ (reminiscent of Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\mathbf{37}$, 1962]); (ii) if a…
We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics…
Using the language of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is then used to obtain an explicit representation of the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.
We study a large class of long-range random walks which take values on the vertices of an N dimensional hypercube. These processes are connected with multivariate Bernoulli autoregression.
We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting behavior of extreme observations until a given time $t$ tends to be rather involved. We describe this asymptotics and generalize several partial results which appeared in this setting. In contrast to the earlier studies, our analysis is based in the point processes theory. The theory is applied to determine the asymptotic distribution of maximal excursions and sojo…
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $α$-stable limit distributions o…
Consider a random walk $S_n=\sum_{i=0}^n X_i$ with negative drift. This paper deals with upper bounds for the maximum $M=\max_{n\ge 1}S_n$ of this random walk in different settings of power moment existences. As it is usual for deriving upper bounds, we truncate summands. Therefore we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby we can reduce the problem of finding upper bounds for $M$ to the problem of finding upper bounds for $M_τ=\max_{n\le τ}S_n$. In addition we test ou…
We consider subordinators $X_α=(X_α(t))_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator $(S_α(t))_{t\ge 0}$ (where $α\in(0,1)$); thus, with the property that $\overlineΠ_α$, the tail function of the canonical measure of $X_α$, is regularly varying of index $-α\in (-1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $α=0$, when $\overlineΠ_α$ is slowly varying at 0. When $α\in(0,1)$, we show that $(t \overlineΠ_α(X_α(t)))^{-1}$ converges in distribution, as $t\downarrow 0$, to the random variable $(S_α(1))^α$. This latter random variable, as a function of $α$, conv…
Let $(M,d,μ)$ be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on $M$ symmetric with respect to $μ$ and whose one-step transition density is comparable to $ (V_h(d(x,y)) ϕ(d(x,y))^{-1}$, where $ϕ$ is a positive continuous regularly varying function with index $β\in (0,2)$ and $V_h$ is the homogeneous volume growth function. Extending several existing work by other authors, we prove global upper and lower bounds for $n$-step transition probability density that are sharp up to constants.
We consider a branching random walk on a multi($Q$)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type $Q$ have regularly varying tails of index $α$, while the other types of particles have lighter tails than that of particles of type $Q$. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the $n^{th}$ generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPP…
Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mm_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than -1. The paper presents sufficient conditions of the almost sure convergence of the series $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish a criterion of finiteness of $\me W\log^+ W a(\log^+W)$ and $\me \log^+|\zi| a(…
We consider the sums $S_n=ξ_1+\cdots+ξ_n$ of independent identically distributed random variables. We do not assume that the $ξ$'s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${\bf P}\{M>x\}$ as $x\to\infty$, provided that $M=\sup\{S_n,\ n\ge1\}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a numbe…
It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wide interest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of $\mathbb{R}^d$-valued Markov processes with the components of an independent multivariate inverse subordinator. As a possible application, we present a model of anomalous diffusion in anisotropic medium, which is obtained as a w…
The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the $N$ rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behav…
