We investigate a stationary random coefficient autoregressive process.
Using renewal type arguments tailor-made for such processes, we show that the stationary distribution has a power-law tail. When the model is normal, we show that the model is in distribution equivalent to an autoregressive process with ARCH errors. Hence, we obtain the tail behavior of any such model of arbitrary order.
In this paper we consider stationary Markov chains with trivial two-sided tail sigma field, and prove that additive functionals satisfy the central limit theorem provided the variance of partial sums divided by n is bounded.
Sticky Brownian motions, as time-changed semimartingale reflecting Brownian motions, have various applications in many fields, including queuing theory and mathematical finance. In this paper, we are concerned about the stationary distributions of a multidimensional sticky Brownian motion, provided it is stable. We will study the large deviations principle for stationary distribution and the tail behaviour of the joint stationary distribution.
We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotics behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on a well-known time-evolution formula of a Markov process, so-called Dynkin's formula, where a key ingredient is a suitable choice of test functions. Those results show how multidimensional tail asymptotics can be studied for the more than two-dimensional ca…
Extremal clusters of stationary processes with long memory can be quite intricate. For certain stationary infinitely divisible processes with subexponential tails, including both power-like tails and certain lighter tails, e.g. lognormal-like tails, such clusters may take the shape of stable regenerative sets. In this paper we show that for semi-exponential tails, which are even lighter, a new shape of extremal clusters arises. In this case each stable regenerative set supports a random panoply of varying extremes.
We present a proof of an upper tail bound of the correct order (up to a constant factor in the exponent) in two classes of stationary models in the KPZ universality class. The proof is based on an exponential identity due to Rains in the case of Last Passage Percolation with exponential weights, and recently re-derived by Emrah-Jianjigian-Seppäiläinen (EJS). Our proof follows very similar lines for the two classes of models we consider, using only general monotonocity and convexity properties, and can thus be expected to apply to many other stationary models.
In this paper, we consider a two-dimensional sticky Brownian motion. Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. For example, a sticky Brownian motion can be used to model a storage system.with exceptional services. In this paper, we focus on stationary distributions for sticky Brownian motions. The main results obtained here include tail asymptotic properties in boundary stationary distributions, marginal distributions, and joint distributions. The ke…
This work is denoted to studying the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime-switching. One essential difference shown in this work between CIR process with regime-switching and without regime-switching is that the stationary distribution for CIR process with regime-switching could be heavy-tailed. Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence. In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with re…
Consider an multidimensional obliquely reflected Brownian motion in the positive orthant, or, more generally, in a convex polyhedral cone. We find sufficient conditions for existence of a stationary distribution and convergence to this distribution at the exponential rate, as time goes to infinity. We also prove that certain exponential moments for this distribution are finite, thus providing a tail estimate for this distribution. Finally, we apply these results to systems of rank-based competing Brownian particles.
We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) $\{(\boldsymbol{X}_n,J_n)\}$ on $\mathbb{Z}_+^2\times S_0$, where $\boldsymbol{X}_n=(X_{1,n},X_{2,n})$ is the level state, $J_n$ the phase state (background state) and $S_0$ a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in an…
Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group $\mathbb{G}$. The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure $ξ$ associated with a stationary (measurable) random field $Y=(Y_s)_{s\in \mathbb{G}}$. It is important to allow the underlying stationary measure to be $σ$-finite. Then we proceed to a random field (defined on a prob…
The semimartingale reflecting Brownian motion (SRBM) can be a heavy traffic limit for many server queueing networks. Asymptotic properties for stationary probabilities of the SRBM have attracted a lot of attention recently. However, many results are obtained only for the two-dimensional SRBM. There is only little work related to higher dimensional ($\geq 3$) SRBMs. In this paper, we consider a three dimensional SRBM: A three dimensional Brownian-driven tandem queue with intermediate inputs. We are interested in tail asymptotics for stationary distributions. By generalizing the kernel method an…
We study moderate deviations in the exponential corner growth model, both in the bulk setting and the increment-stationary setting. The main results are sharp right-tail bounds on the last-passage time and the exit point of the increment-stationary process. The arguments utilize calculations with the stationary version and a moment generating function identity due to E. Rains, for which we give a short probabilistic proof. As applications of the deviation bounds, we derive upper bounds on the speed of distributional convergence in the Busemann function and competition interface limits.
A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigrati…
We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two station case, this problem has been recently solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probab…
We investigate asymptotics of the tail distribution of sojourn time $$ \int_0^T \mathbb{I}(X(t)> u)dt, $$ as $u\to\infty$, where $X$ is a centered stationary Gaussian process and $T$ is an independent of $X$ nonnegative random variable. The heaviness of the tail distribution of $T$ impacts the form of the asymptotics, leading to four scenarios: the case of integrable $T$, the case of regularly varying $T$ with index $λ=1$ and index $λ\in(0,1)$ and the case of slowly varying tail distribution of $T$. The derived findings are illustrated by the analysis of the class of fractional Ornstein-Uhlenb…
In this paper, we provide a review on the kernel method, which is one of the options for characterizing so-called exact tail asymptotic properties in stationary probabilities of two-dimensional random walks, discrete or continuous (or mixed), in the quarter plane. Many two-dimensional queueing systems can be modelled via these types of random walks. Stationary probabilities are one of the most sought statistical quantities in queueing analysis. However, explicit expressions are available only for a very limited number of models. Therefore, tail asymptotic properties become more important, sinc…
The tail process $\boldsymbol{Y}=(Y_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ of a stationary regularly varying random field $\boldsymbol{X}=(X_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ represents the asymptotic local distribution of $\boldsymbol{X}$ as seen from its typical exceedance over a threshold $u$ as $u\to\infty$. Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedance-stationarity and that this property, together with the polar decomposition of the tail process, characterizes the class of all tail pro…
In this paper, we investigate exact tail asymptotics for the stationary distribution of a fluid model driven by the $M/M/c$ queue, which is a two-dimensional queueing system with a discrete phase and a continuous level. We extend the kernel method to study tail asymptotics of its stationary distribution, and a total of three types of exact tail asymptotics is identified from our study and reported in the paper.
In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Aza\"ıs and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Aza\"ıs and Wschebor that shows that under some conditi…
We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $ρ\in (0,1)$ and points $v_N=((1-ρ)^2 N,ρ^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $ρ=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on …
We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) on $\mathbb{Z}_+^2\times S_0$, where $S_0$ is a finite set, and give a complete expression for the asymptotic decay function of the stationary tail probabilities in an arbitrary direction. The 2d-QBD process is a kind of random walk in the quarter plane with a background process. In our previous paper (Queueing Systems, vol. 102, pp. 227-267, 2022), we have obtained the asymptotic decay rate of the stationary tail probabilities in an arbitrary direction and clarified that if the asymptotic dec…
The sums and maxima of weighted non-stationary random length sequences of regularly varying random variables may have the same tail and extremal indices, Markovich and Rodionov (2020). The main constraints are that there exists a unique series in a scheme of series with the minimum tail index, the tail of the term number is lighter than the tail of the terms and the weights are positive constants. These assumptions are changed here: a bounded random number of series is allowed to have the minimum tail index, the tail of the term number may be heavier than the tail of the terms and the weights …
We propose a discrete-time, finite-state stationary process that can possess long-range dependence. Among the interesting features of this process is that each state can have different long-term dependency, i.e., the indicator sequence can have different Hurst index for different states. Also, inter-arrival time for each state follows heavy tail distribution, with different states showing different tail behavior. A possible application of this process is to model over-dispersed multinomial distribution. In particular, we define fractional multinomial distribution from our model.
We provide a generalisation of Pinelis' Rademacher-Gaussian tail comparison to complex coefficients. We also establish uniform bounds on the probability that the magnitude of weighted sums of independent random vectors uniform on Euclidean spheres with matrix coefficients exceeds its second moment.
Given that a stationary Gaussian process is above a high threshold, the length of time it spends before going below that threshold is studied. The asymptotic order is determined by the smoothness of the sample paths, which in turn is a function of the tails of the spectral measure. Two disjoint regimes are studied - one in which the second spectral moment is finite and the other in which the tails of the spectral measure are regularly varying and the second moment is infinite.
We study clustering of the extremes in a stationary sequence with subexponential tails in the maximum domain of attraction of the Gumbel We obtain functional limit theorems in the space of random sup-measures and in the space $D(0,\infty)$. The limits have the Gumbel distribution if the memory is only moderately long. However, as our results demonstrate rather strikingly, the "heuristic of a single big jump" could fail even in a moderately long range dependence setting. As the tails become lighter, the extremal behavior of a stationary process may depend on multiple large values of the driving…
We investigate maxima of linear processes with i.i.d. heavy-tailed innovations and random coefficients. Using the point process approach we derive functional convergence of the partial maxima stochastic process in the space of non-decreasing càdlàg functions on $[0,1]$ with the Skorohod $M_{1}$ topology.
We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we obtain functional convergence of the corresponding partial sum stochastic process in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology.
Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients
LMAM, School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
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.
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $\mathbb{R}^{2}$-valued càdlàg functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
