Paolo Baldi
We prove a simple criterion of exponential tightness for sequences of Gaussian r.v.'s with values in a separable Banach space from which we deduce a general result of Large Deviations which allows easily to obtain LD estimates in various situations.
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Let us consider a Gaussian probability on a Banach space. We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS. Such a space has full probability and a compact embedding. This extends what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result it is very simple to deduce a result of exponential tightness for Gaussian probabilities.
On tightness and exponential tightness in generalised Jackson networks
本文通过统一证明，探讨了广义杰克逊网络中平稳队列长度序列的紧性与指数紧性，涵盖大偏差、正常偏差和中等偏差等多种情形。
Exponential tightness of a family of Skorohod integrals is studied in this paper. We first provide a counterexample to illustrate that in general the exponential tightness with speed $ε$ similar to Itô integral does not hold, even for any speed $ε^α$ with $α> 0$. Then, some characterizations of this subject are given. Application is also provided to illustrate the proposed results.
We formulate and prove in this report some sufficient conditions for exponential tightness (ET) of a family of independent identical distributed (i.i.d.) random fields (r.f.) (processes) in the space of continuous functions defined on certain metric compact set. We will use the entropic method as well as too modern approach through the majorizing measures (generic chaining).
Let $Φ'_β$ denotes the strong dual of a nuclear space $Φ$ and let $D_{T}(Φ'_β)$ be the Skorokhod space of right-continuous with left limits (càdlàg) functions from $[0,T]$ into $Φ'_β$. In this article we introduce the concepts of cylindrical random variables and cylindrical measures on $D_{T}(Φ'_β)$, and prove analogues of the regularization theorem and Minlos theorem for extensions of these objects to bona fide random variables and probability measures on $D_{T}(Φ'_β)$ respectively. Later, we establish analogues of Lévy's continuity theorem to provide necessary and sufficient conditions for u…
We consider a family of centered Gaussian fields on the d-dimensional unit box, whose covariance decreases logarithmically in the distance between points. We prove tightness of the recentered maximum of the Gaussian fields and provide exponentially decaying bounds on the right and left tails. We then apply this result to a version of the two-dimensional continuous Gaussian free field.
In this paper we introduce a framework to prove tightness of a sequence of discrete Gibbsian line ensembles $\mathcal{L}^N = \{\mathcal{L}_k^N(x), k \in \mathbb{N}, x \in \frac{1}{N}\mathbb{Z}\}$, which is a collection of countable random curves. The sequence of discrete line ensembles $\mathcal{L}^N$ we consider enjoys a resampling invariance property, which we call $(H^N,H^{RW,N})$-Gibbs property. We also assume that $\mathcal{L}^N$ satisfies technical assumptions A1-A4 on $(H^N,H^{RW,N})$ and the assumption that the lowest labeled curve with a parabolic shift, $\mathcal{L}_1^N(x) + \frac{x^…
In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded SDEs by the Arzelà--Ascoli tightness characterization of the space $\C$ equipped with the uniform topology (b) neutral SDEs with continuous sample paths by a generalized Razumikhin-type argument and a stability-in-distribution approach and (c) jump-d…
We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $Λ(\cdot)_{\mid \mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when …
We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us to get precise estimates on boundary connection probabilities. It remains purely discrete, in particular we do not make use of any continuum limit, and it can be used to derive directly several noteworthy results - some new and some not - among which the fact that there is no spontaneous magnetization at criticality, tightness properties for the …
Nonequilibrium moderate deviations from hydrodynamics of simple exclusion processes
本文证明了从非平衡态出发的一维环上简单排他过程流体动力学极限的中等偏差原理，扩展了平衡态结果，利用指数紧性和替换引理及Gronwall不等式等工具。\n
Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.
Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension
研究\(\mathbb{R}^d\)（\(d\geq2\)）上类似刘维尔量子引力度量的自然近似方案，通过特定高斯随机函数序列构建随机度量族，证明在亚临界阶段经重标度后度量紧致，子序列极限诱导欧几里得拓扑，并列出诸多开放问题。
We explicitly find the rate of exponential long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows to reduce the pro blem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.
We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an $o(\log n)$ bound by Chatterjee (2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (2010), who proved the tightness of the centered maximum together with an evaluati…
Large Deviations for Stochastic Differential Equations Driven by Semimartingales
本文研究由半鞅驱动的随机微分方程，利用驱动半鞅的特征给出条件，在噪声-控制对满足良好速率函数的大偏差原理时，证明解过程也具有大偏差原理，且无需联合或一致指数紧性假设。
Consider the 3D Ising model on a box of side length $n$ with minus boundary conditions above the $xy$-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height $M_n$ of this interface: for every $β$ large, $M_n/ \log n\to c_β$ in probability as $n\to\infty$. Here we show that the laws of the centered maxima $(M_n - \mathbb{E}[M_n])_{n\geq…
This is the first of a three paper series in which we present a comprehensive study of the extreme value theory of the scale-inhomogeneous discrete Gaussian free field. This model was introduced by Arguin and Ouimet who computed the first order of the maximum. In this first paper we establish tail estimates for the maximum value, which allow to deduce the log-correction to the order of the maximum and tightness of the centred maximum. Our proofs are based on the second moment method and Gaussian comparison techniques.
In this paper we present various new inequalities for tail proabilities for distributions that are elements of the most improtant exponential families. These families include the Poisson distributions, the Gamma distributions, the binomial distributions, the negative binomial distributions and the inverse Gaussian distributions. All these exponential families have simple variance functions and the variance functions play an important role in the exposition. All the inequalities presented in this paper are formulated in terms of the signed log-likelihood. The inequalities are of a qualitative n…
We prove the exponential ergodicity of the transition probabilities of solutions to elliptic multivalued stochastic differential equations.
We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.
We analyze here different forms of fractional relaxation equations of order ν\in(0,1) and we derive their solutions both in analytical and in probabilistic forms. In particular we show that these solutions can be expressed as crossing probabilities of random boundaries by various types of stochastic processes, which are all related to the Brownian motion B. In the special case ν=1/2, the fractional relaxation is proved to coincide with Pr{sup_{0\leqs\leqt} B(s)<U}, for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the Ga…
In this paper we show irreducibility and the strong Feller property for transition probabilities of stochastic differential equations with jumps and monotone coefficients. Thus, exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained.
Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.
Upper exponential inequalities for the tail probabilities of the centered and normalized number of triangles in the Erdös-Rényi graph are obtained, where the probability of every edge is fixed. The result is formulated in terms of random fields.
Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay faster than exponentially. It is shown that if the spectral measure is not singular, then the exponent in the persistence probability cannot grow faster than quadratically. An example that appears (from numerical evidence) to achieve this lower bound is presented.
Poisson genericity in numeration systems with exponentially mixing probabilities
本文为具有不变指数混合概率测度的有限或可数字母表序列定义了 Poisson 通用性，并证明几乎所有序列均满足该性质，推广了 Peres 和 Weiss 的定理，得出几乎所有实数的连分数展开也是 Poisson 通用的。
We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, $\mathbb{P}_p[0\text{ connected to }\infty]\geq c(p-p_c)$. For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.
We prove the super-exponential decay of probabilities that there exist $n$ crossings of a given quadrilateral in a simple $\text{CLE}_κ(Ω)$, $\frac{8}{3}<κ\le 4$, as $n$ goes to infinity. Besides being of independent interest, this also provides the missing ingredient in arXiv:1809.00690 for proving the convergence of probabilities of cylindrical events for the double-dimer loop ensembles to those for the nested $\text{CLE}_4(Ω)$.
Let $\{(X(t), Y(s)): t\in T, s\in S\}$ be an $\mathbb{R}^2$-valued, centered, unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint excursion probability $\mathbb{P} \{\sup_{t\in T} X(t) \geq u, \sup_{s\in S} Y(s) \geq u \}$ can be approximated by $\mathbb{E}\{χ(A_u)\}$, the expected Euler characteristic of the excursion set $A_u=\{(t,s)\in T\times S: X(t) \ge u, Y(s) \ge u\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic (cf. Taylor, Ta…
Rare event probabilities in Random Geometric Graphs
本文研究高维球面和高斯随机几何图中的稀有事件，通过几何与概率论证，获得图完全性与异常大边数概率的渐近指数衰减率。
We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in the M/G/1 queue, pricing of barrier options, etc). On the way, we provide a new approximation, for the perturbed Cramer-Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe, which fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whi…
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose sev…
Persistence probabilities of MA(1) sequences with Laplace innovations and $q$-deformed zigzag numbers
本文研究了一阶移动平均序列在Laplace创新下的持久概率，利用q-变形的组合量和生成函数进行显式计算。
In this paper, we adapt the classic Cramér-Lundberg collective risk theory model to a perturbed model by adding a Wiener process to the compound Poisson process, which can be used to incorporate premium income uncertainty, interest rate fluctuations and changes in the number of policyholders. Our study is part of a Master dissertation, our aim is to make a short overview and present additionally some new approximation methods for the infinite time ruin probabilities for the perturbed risk model. We present four different approximation methods for the perturbed risk model. The first method is b…
A sharp transition in zero overcrowding and undercrowding probabilities for Stationary Gaussian Processes
本文研究平稳高斯过程零点数量的过密与欠密概率，证明当谱测度支撑在 $\pm[B,A]$ 时，两类概率分别在 $η=\tfrac{A}π$ 和 $η=\tfrac{B}π$ 处发生指数衰减与高斯衰减之间的急剧转变。
Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$ and $\limsup_{n\to\infty} p_c(G_n) < 1$, then $ p_c(G_n)\to p_c(G)$ as $n\to\infty$. We verify this conjecture under the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable. In the unimodular case, this res…
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