Large Deviations for Hierarchical Systems of Interacting Jump Processes

We investigate the large deviations principle from the McKean–Vlasov limit for a collection of jump processes obeying a two-level hierarchy interaction. A large deviation upper bound is derived and it is shown that the associated rate function admits a Lagrangian representation as well as a nonvariational one. Moreover, it is proved that the admissible paths for the weak solution of the McKean–Vlasov equation enjoy certain strong differentiability properties.     

Large and Moderate Deviations for Estimators of Quadratic Variational Processes of Diffusions

For a diffusion process dX_t = σdB _t + b(t, X_t)dt with (σ _t ) unknown, we study the large and moderate deviations of the estimator of the quadratic variational process . This revised version was published online in June 2006 with corrections to the Cover Date.     

Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions

Journal of Theoretical Probability - Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an...     

On the McKean-Vlasov Limit for Interacting Diffusions

For a system of diffusions in a domain of R^d with long-range weak interaction the behavior of the associated empirical process is studied. Under mild growth and smoothness assumptions on the drift and diffusion coefficients such as coercivity and monotonicity conditions the law of large numbers and the propagation of chaos are proved. Existence and uniqueness of the weak solution to the McKean - Vlasov equation and the associated non-linear martingale problem are investigated.     

On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

Large Deviations for Quicksort

LetQ_nbe the random number of comparisons made by quicksort in sortingndistinct keys when we assume that alln! possible orderings are equally likely. Known results concerning moments forQ_ndo not show how rare it is forQ_nto make large deviations from its mean. Here we give a good approximation to the probability of such a large deviation and find that this probability is quite small. As well as the basic quicksort, we consider the variant in which the partitioning key is chosen as the median of (2t+1) keys.     

Moderate Deviations and Large Deviations for Kernel Density Estimators

Let f _n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ^d . It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) 0 as |x| , then the moderate deviation principle and large deviation principle for hold.     

Precise Large Deviations for Long-Tailed Distributions

In this paper, we investigate the precise large deviations for sums of independent identically distributed random variables with heavy-tailed distributions. We prove asymptotic relations for non-random sums and for random sums of random variables with long-tailed distributions. We apply the results on two useful counting processes, namely, renewal and compound-renewal processes.     

Large Deviations for Some Dependent Sequences

Let （Xi） be a martingale difference sequence and Sn=∑^ni=1Xi Suppose （Xi） i=1 is bounded in L^p. In the case p ≥2, Lesigne and Volny （Stochastic Process. Appl. 96 （2001） 143） obtained the estimation μ（Sn 〉 n） ≤ cn^-p/2, Yulin Li （Statist. Probab. Lett. 62 （2003） 317） generalized the result to the case when p ∈ （1,2] and obtained μ（Sn 〉 n） ≤ cn^l-p, these are optimal in a certain sense. In this article, the authors study the large deviation of Sn for some dependent sequences and obtain the same order optimal upper bounds for μ（Sn 〉 n） as those for martingale difference sequence.     

Large Deviations for Symmetrised Empirical Measures

In this paper we prove a Large Deviation Principle for the sequence of symmetrised empirical measures \(\frac{1}{n}\sum_{i=1}^{n}\delta_{(X^{n}_{i},X^{n}_{\sigma_{n}(i)})}\) where σ _n is a random permutation and ((X _i ⁿ )_1≤i≤n ) _n≥1 is a triangular array of random variables with suitable properties. As an application we show how this result allows to improve the Large Deviation Principles for symmetrised initial-terminal conditions bridge processes recently established by Adams, Dorlas and König.     

Large Deviations for Brownian Intersection Measures

We consider p independent Brownian motions in \input amssym ${\Bbb R}^d$ . We assume that p ≥ 2 and p (d ? 2) < d. Let ?_t denote the intersection measure of the p paths by time t, i.e., the random measure on \input amssym ${\Bbb R}^d$ that assigns to any measurable set \input amssym $A \subset {\Bbb R}^d$ the amount of intersection local time of the motions spent in A by time t. Earlier results of X. Chen derived the logarithmic asymptotics of the upper tails of the total mass \input amssym $\ell _t \left({{\Bbb R}^d } \right)$ as t → ∞. In this paper, we derive a large‐deviation principle for the normalized intersection measure t^?p?_t on the set of positive measures on some open bounded set \input amssym $B \subset {\Bbb R}^d$ as t → ∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalized occupation times measures of the p motions. Our proof makes the classical Donsker‐Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set . This extends earlier studies on the intersection measure by König and Mörters. © 2012 Wiley Periodicals, Inc.     

Tanaka Formula and Local Time for a Class of Interacting Branching Measure-valued Diffusions

We construct superprocesses with dependent spatial motion(SDSMs) in Euclidean spaces R^d with d≥1 and show that,even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on R^d,their local times exist when d≤3.A Tanaka formula of the local time is also derived.     

Large Deviations for Multivalued Stochastic Differential Equations

We prove a large deviation principle of Freidlin–Wentzell type for multivalued stochastic differential equations with monotone drifts that in particular contain a class of SDEs with reflection in a convex domain.     

Large Deviations Rate Function for Polling Systems

In this paper, we identify the local rate function governing the sample path large deviation principle for a rescaled process n ^–1 Q _nt , where Q _t represents the joint number of clients at time t in a polling system with N nodes, one server and Markovian routing. By the way, the large deviation principle is proved and the rate function is shown to have the form conjectured by Dupuis and Ellis. We introduce a so called empirical generator consisting of Q _t and of two empirical measures associated with S _t , the position of the server at time t. One of the main step is to derive large deviations bounds for a localized version of the empirical generator. The analysis relies on a suitable change of measure and on a representation of fluid limits for polling systems. Finally, the rate function is solution of a meaningful convex program. The method seems to have a wide range of application including the famous Jackson networks, as shown at the end of this study. An example illustrates how this technique can be used to estimate stationary probability decay rate.     

Large Deviations for Intersections of Random Walks

We prove a large deviations principle for the number of intersections of two independent infinite-time ranges in dimension 5 and greater, improving upon the moment bounds of Khanin, Mazel, Shlosman, and Sinaï [9]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen, and den Hollander [15], who analyzed this question for the Wiener sausage in the finite-time horizon. The proof builds on their result (which was adapted in the discrete setting by Phetpradap [12]), and combines it with a series of tools that were developed in recent works of the authors [2, 3, 5]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order 1. © 2022 Wiley Periodicals, Inc.     

经验测度序列的大偏差原理

设{Xi,I∈N)是平稳NA随机变量序列且ψ-(1)＞0.记经验测度δn=1/n∑I=1δxi,n≥1,借助于弱收敛拓扑下的开集与β度量下的开球之间的关系,证明了{P{δn∈·},n→∞}在(M1(R),ω→)上满足大偏差原理.     

Upper Large Deviations for Mixing Random Sequence

In this article, we prove upper large deviations for the empirical measure generated by stationary mixing random sequence under some suitable assumptions and upper large deviations for the mixing random sequence.