Exact Asymptotics Research Articles

Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

Let \( \mathcal{T} \) be a positive random variable independent of a real-valued stochastic process \( \left\{ {X(t),t\geqslant 0} \right\} \). In this paper, we investigate the asymptotic behavior of \( \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right) \) as u→∞ assuming that X is a strongly dependent stationary Gaussian process and \( \mathcal{T} \) has a regularly varying survival function at infinity with index λ ∈ [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that \( \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right)={c^{\lambda }}\mathrm{P}\left( {\mathcal{T}>m(u)} \right)\left( {1+o(1)} \right) \) with some positive finite constant c and function m(·) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution.

Exact tail asymptotics of aggregated parametrised risk

In this paper we investigate the extremal behaviour of aggregated risk for a specific parametrised multivariate dependence framework. Furthermore we discuss conditional limit results and extremal behaviour of both maximum and aggregated log-elliptical risk. Our application establishes the logarithmic efficiency of the Rojas-Nandaypa algorithm for rare-event simulation of log-elliptical risks.

Wireless three-hop networks with stealing II: exact solutions through boundary value problems

We study the stationary distribution of a random walk in the quarter plane arising in the study of three-hop wireless networks with stealing. Our motivation is to find exact tail asymptotics (beyond logarithmic estimates) for the marginal distributions, which requires an exact solution for the bivariate generating function describing the stationary distribution. This exact solution is determined via the theory of boundary value problems. Although this is a classical approach, the present random walk exhibits some salient features. In fact, to determine the exact tail asymptotics, the random walk presents several unprecedented challenges related to conformal mappings and analytic continuation. We address these challenges by formulating a boundary value problem different from the one usually seen in the literature.

Behavior at infinity of a solution to a differential-difference equation

We obtain an asymptotic expansion for a solution to a nonhomogeneous retarded- or neutraltype differential-difference equation. The case of unbounded delays is considered. The influence is accounted for the roots of the characteristic equation. We establish the exact asymptotics for the remainder depending on the asymptotic properties of the free matrix term of the equation.

Exact tail asymptotics for the M/M/m retrial queue with nonpersistent customers

We consider the M/M/m retrial queue with nonpersistent customers. Liu et al. (2011) [12] provided the asymptotic lower and upper bounds for the stationary distribution of the number of customers in the orbit. In this paper we strengthen Liu, Wang and Zhao’s result by finding the exact tail asymptotic formula.

Cauchy problem for a parabolic pseudodifferential higher-order equation with pulse action

UDC 517.956 We consider the Cauchy problem and the problem with pulse action for a pseudodifferential higherorder equation with respect to t . We construct solutions of these problems, study their properties, and prove a theorem on correctness. The theory of pseudodifferential operators and pseudodifferential equations, whose modern form was formed in the middle of the 1960s, is studied in many works [2–8, 11, 12]. After determination that pseudodifferential operators are closely connected with problems of analysis and contemporary mathematical physics, especially in the theory of elliptic boundary-value problems [10], the number of such works has considerably increased. Linear parabolic pseudodifferential equations with nonsmooth symbols were defined by Eidel’man, Drin’, and Iwasaki at the beginning of the 1970s in [2, 3, 8, 9, 11, 12]. The symbols of these pseudodifferential operators are nonsmooth at the point σ= 0 , σ∈ R n . For this reason, standard methods, which are used for pseudodifferential operators with smooth symbols, cannot be applied to the investigation of problems for these pseudodifferential equations. Investigation of these equations with constant (independent of the space coordinates x ∈R n and the time coordinate t ∈(0,T ]) homogeneous symbol was originated in [8]. The fundamental solution of the Cauchy problem for these equations was determined with the use of the Fourier transformation. In [7], Fedoryuk establishes that the exact asymptotics of the fundamental solution of the Cauchy problem as x →∞ is power rather than exponential as for differential equations. In [3], Schauder estimates are obtained and the correct solvability of the Cauchy problem in classes of Holder functions is established. For the subsequent development of this theory, Kochubei’s works [4, 11] were very important. In these works, for the first time, it is noted that pseudodifferential operators with nonsmooth symbols can be interpreted as hypersingular integral operations. This enabled one to use the well-developed theory of hypersingular integral operations for the investigation of the Cauchy problem. Kochubei constructed and studied fundamental solutions of the Cauchy problem, proved theorems on solvability of the Cauchy problem in classes of functions with power increase as x →∞ , and indicated connections of the obtained results with theory of random processes.

Analysis of exact tail asymptotics for singular random walks in the quarter plane

In this paper, we consider all singular cases of random walks in the quarter plane. Specifically, exact light tail asymptotics for stationary probabilities are obtained for all singular random walks.

Integral functionals for the exponential of the wiener process and the Brownian bridge: Exact asymptotics and legendre functions

We prove results concerning the exact asymptotics of the probabilities $$P\left\{ {\int_0^1 {e^{\varepsilon \xi (t)} dt < b} } \right\}, P\left\{ {\int_0^1 {e^{\varepsilon \left| {\xi (t)} \right|} dt < b} } \right\}$$ as ɛ → 0 and 0 < b < 1 for two Gaussian processes ξ(t), the Wiener process and the Brownian bridge. We also obtain asymptotic formulas for integrals of Laplace type. Our study is based on the Laplace method for Gaussian measures in Banach spaces. The calculations of the constants are reduced to the solution of an extremal problem for the action functional and to the study of the spectrum of a second-order differential operator of Sturm-Liouville type using the Legendre functions.

Negative-order moments for -functionals of Wiener processes: exact asymptotics

We prove theorems on the exact asymptotics as of the integrals , , for two stochastic processes , the Wiener process and the Brownian bridge, as well as for their conditional versions. We also obtain a number of related results. We shall use the Laplace method for the occupation times of homogeneous Markov processes. We write the constants in our exact asymptotic formulae explicitly in terms of the minimal eigenvalue and corresponding eigenfunction for the Schrödinger operator with a potential of polynomial type.

On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials

The asymptotic behavior of solutions to Schrodinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, pointwise upper estimate are derived.

Mismatch of conductivity anisotropy in the mixed and normal states of type-II superconductors

We have calculated the Bardeen-Stephen contribution to the vortex viscosity for uniaxial anisotropic superconductors within the time-dependent Ginzburg-Landau (TDGL) theory. We focus our attention on superconductors with a mismatch of anisotropy of normal and superconducting characteristics. Exact asymptotics for the Bardeen-Stephen contribution have been derived in two limits: the cases of small and large electric field penetration depth (as compared to the coherence length). Also we suggest a variational procedure which allows us to calculate the vortex viscosity for superconductors with arbitrary ratio of the coherence lenght to the electric field penetration depth. The approximate analytical result is compared with numerical calculations. Finally, using a generalized TDGL theory, we prove that the viscosity anisotropy and, thus, the flux-flow conductivity anisotropy may depend on temperature.

Sequential estimation of a threshold crossing time for a Gaussian random walk through correlated observations

Given a Gaussian random walk X with drift, we consider the problem of estimating its first-passage time ? A for a given level A from an observation process Y correlated to X. Estimators may be any stopping times ? with respect to the observation process Y. Two cases of the process Y are considered: a noisy version of X and a process X with delay d. For a given loss function f(x), in both cases we find exact asymptotics of the minimal possible risk E f((? ? ? A )/r) as A, d ? ?, where r is a normalizing coefficient. The results are extended to the corresponding continuous-time setting where X and Y are Brownian motions with drift.

Some exact asymptotics in the counting of walks in the quarter plane

Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). The aim of this paper is to propose a new approach to obtain some exact asymptotics for walks confined to the quarter plane.

On the measure of conformal difference between Euclidean and Lobachevsky spaces

Euclidean space and Lobachevsky space are known to be not equivalent either conformally or quasiconformally. In this work we give exact asymptotics of the critical order of growth at infinity for the quasiconformality coefficient of a diffeomorphism for which such a mapping is possible. We also consider the general case of immersions of conformally parabolic Riemannian manifolds.Bibliography: 17 titles.

Limits on Support Recovery of Sparse Signals via Multiple-Access Communication Techniques

In this paper, we consider the problem of exact support recovery of sparse signals via noisy linear measurements. The main focus is finding the sufficient and necessary condition on the number of measurements for support recovery to be reliable. By drawing an analogy between the problem of support recovery and the problem of channel coding over the Gaussian multiple-access channel (MAC), and exploiting mathematical tools developed for the latter problem, we obtain an information-theoretic framework for analyzing the performance limits of support recovery. Specifically, when the number of nonzero entries of the sparse signal is held fixed, the exact asymptotics on the number of measurements sufficient and necessary for support recovery is characterized. In addition, we show that the proposed methodology can deal with a variety of models of sparse signal recovery, hence demonstrating its potential as an effective analytical tool.

On the Haezendonck–Goovaerts risk measure for extreme risks

In this paper, we are interested in the calculation of the Haezendonck–Goovaerts risk measure, which is defined via a convex Young function and a parameter q ∈ ( 0 , 1 ) representing the confidence level. We mainly focus on the case in which the risk variable follows a distribution function from a max-domain of attraction. For this case, we restrict the Young function to be a power function and we derive exact asymptotics for the Haezendonck–Goovaerts risk measure as q ↑ 1 . As a subsidiary, we also consider the case with an exponentially distributed risk variable and a general Young function, and we obtain an analytical expression for the Haezendonck–Goovaerts risk measure.

Exact asymptotics of supremum of a stationary Gaussian process over a random interval

Let { X ( t ) : t ∈ [ 0 , ∞ ) } be a centered stationary Gaussian process. We study the exact asymptotics of P ( sup s ∈ [ 0 , T ] X ( s ) > u ) , as u → ∞ , where T is an independent of { X ( t ) } nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T , the case of T having regularly varying tail distribution with parameter λ ∈ ( 0 , 1 ) and the case of T having slowly varying tail distribution.

Exact Asymptotics of the Freezing Transition of a Logarithmically Correlated Random Energy Model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation—thus translating Bramson’s work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point.

Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method

We prove results on exact asymptotics as for the expectations and probabilities , where is a sequence of independent identically Laplace-distributed random variables, , , is the corresponding random walk on , is a positive continuous function satisfying certain conditions, and , , are fixed numbers. Our results are obtained using a new method which is developed in this paper: the Laplace method for the occupation time of discrete-time Markov chains. For one can take , , , , or , , , for example. We give a detailed treatment of the case when using Bessel functions to make explicit calculations.

Asymptotic variance of the self-intersections of stable random walks using Darboux-Wiener theory

We present a Darboux-Wiener type lemma as a powerful alternative to the classical Tauberian theorem when monotonicity is not known a priori. We apply it to obtain the exact asymptotics of the variance of the self-intersections of a one-dimensional stable random walk. Finally we prove a functional central limit theorem for stable random walk in random scenery conjectured in [1].