The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.
We derive new upper and lower bounds for probabilities that $r$ or at least $r$ from $n$ events occur. These bounds can turn to equalities. The method is discussed as well. It works for measurable space and measures with sign, too. We also discuss variants of the results for conditional probability of above events given $σ$-field. Taking expectations from both parts of inequalities for conditional probabilities can yield better bounds for unconditional ones.
We obtain new lower and upper bounds for probabilities of unions of events.These bounds are sharp. They are stronger than earlier ones. General bounds maybe applied in arbitrary measurable spaces.We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second part of the Borel--Cantelli lemma.
Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. We prove a lower bound of order $n^{- d^2}$ for point-to-point connection probabilities, where $n$ is the distance between the points. Most of the work in our proof concerns a `construction' which finally reduces the problem to a topological one. This is then solved by applying a topological fact, which follows from Brouwer's fixed point theorem. Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in …
We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers.
We consider a system of $d$ non-linear stochastic fractional heat equations in spatial dimension $1$ driven by multiplicative $d$-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of $(u(s, y), u (t, x))$. From this result, we deduce optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualar…
We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $γ_Z$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^p$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case in terms of Newtonian capacity, and imp…
Given a branching random walk$\{Z_n\}_{n\geq0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)$ at generation $n$. It is known from \cite{Biggins1977} that under some mild conditions, $n^{-1}\log Z_n([θx^* n,\infty))$ converges a.s. to $\log m-I(θx^*)$, where $\log m-I(θx^*)$ is a positive constant. In this work, we investigate its lower deviation, in other words, the convergence rates of $$\mathbb{P}\left(Z_n([θx^* n,\infty))<e^{an}\right),$$ where $a\in[0,\log m-I(θx^*))$. Our results complete those in \cite{Mehmet}, \cite{Helower} and \cite{GWlower}…
Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This note describes similar inequalities when, instead of a single measure, the functions are integrated with respect to different measures that form a weakly convergent sequence.
Credal sets are one of the most important models for describing probabilistic uncertainty. They usually arise as convex sets of probabilistic models compatible with judgments provided in terms of coherent lower previsions or more specific models such as coherent lower probabilities or probability intervals. In finite spaces, credal sets usually take the form of convex polytopes. Many properties of convex polytopes can be derived from their normal cones, which form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of coherent lower p…
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.
This paper deals with (finite or infinite) sequences of arbitrary independent events in some probability space. We find sharp lower bounds for the probability of a union of such events when the sum of their probabilities is given. The results have parallel meanings in terms of infinite series.
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.
Given a super-critical branching random walk on $\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2θ^*}\log n$ converges in law for some suitable constants $x^*$ and $θ^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2θ^*}\log n-\ell_n\right),$$ for any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and $\ell_n\uparrow\infty$. As a by-…
We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cramér and Bahadur-Ranga Rao.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
We show bounds on tail probabilities for quadratic forms in sub-gaussian non-necessarily independent random variables. Our main tool will be estimates of the Luxemburg norms of such forms. This will allow us to formulate the above-mentioned bounds. As an example we give estimates of the excess loss in fixed design linear regression in dependent observations.
Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad assumptions on the nature of the boundary, including the Lipschitz condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an analogue of the Borovkov and Novikov (2005) upper bound for the difference between boundary hitting probabilities for "close boundaries" in the univariate case. We also obtained upper bounds for the first boundary crossin…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense that we consider lower expectations (or sets of probabilities) instead of classical 'precise' probabilities. Secondly, instead of binary sequences, we consider sequences whose elements take values in some finite sample space. Interestingly, we find that every sequence is computably random with respect to at least one lower expectation, and that lower expect…
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 …
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…
We consider metric graph Gaussian free field (GFF) defined on polygons of $δ\mathbb{Z}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary data is well-chosen, the scaling limits of crossing probabilities can be explicitly constructed as "fusion" of multiple SLE$_4$ pure partition functions.
We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by some arbitrary compact separable metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are two-term approximations (up to $o(y)$). The asymptotics are formulated in terms of the solution $\tilde f$ to the constrained optimization problem $$ \|h\|_{…
We describe Monte Carlo methods for estimating lower envelopes of expectations of real random variables. We prove that the estimation bias is negative and that its absolute value shrinks with increasing sample size. We discuss fairly practical techniques for proving strong consistency of the estimators and use these to prove the consistency of an example in the literature. We also provide an example where there is no consistency.
Non-existence probabilities and lower tails in the critical regime via Belief Propagation
King’s College London（伦敦国王学院）；Georgia Institute of Technology（佐治亚理工学院）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）；Massachusetts Institute of Technology（麻省理工学院）
We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $γ$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.
We study the limit behaviour of upper and lower bounds on expected time averages in imprecise Markov chains; a generalised type of Markov chain where the local dynamics, traditionally characterised by transition probabilities, are now represented by sets of 'plausible' transition probabilities. Our main result is a set of necessary and sufficient conditions under which these upper and lower bounds, called upper and lower expected time averages, will converge as time progresses towards infinity to limit values that do not depend on the process' initial state. Remarkably, our conditions are cons…
Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices
We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute and are tight within a constant factor of $89/44$. Moreover, they are asymptotically tight in the regimes of large deviation and moderate deviation. By virtue of a surprising connection with Ramanujan's equation, we also provide strong evidences suggesting that the lower bound is tight within a factor of $1.26434$. It may even be regarded as the natural lowe…
One-arm probabilities for the two-dimensional metric-graph and discrete Gaussian free field
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
We consider a $d$-dimensional random field $u=(u(x), x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D\subset \mathbb{R}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$ and its probability law are proved. For Gaussian solutions, using results from [Dalang and Sanz-Solé, 2009], we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel-Riesz capacity, respectively. This relies on precise estimates on the canonical distance of the process or, equivalently, on $L^2$ estimates of inc…
A new formula for the probability that a standard Brownian motion stays between two linear boundaries is proved. A simple algorithm is deduced. Uniform precision estimates are computed. Different implementations have been made available online as R packages.
We provide a lower bound on the expected discovery time of one-way neighbor discovery in multi antenna system. We assume that the scheme allows to use different directional transmission probabilities, which result in different probability of successful discovery between nodes. The proof shows that the lower bound to the expected discovery time, that node $i$ takes to discover its neighbors, is given by the reciprocal of the average probabilities of successful discovery.
Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal H} \leq L$ are satisfied for all times $t \geq 0$, then there is a constant $c = c(L)$ such that for $1 \leq R \leq \sqrt{t}$, \[\mathbb{P}(\|X_t\|_{\mathcal H} \leq R \mid X_0 = x_0) \leq c \frac{R}{\sqrt{t}} e^{-\|x_0\|_{\mathcal H}^2/(6 L^2 t)}\,.\] Following [Lee-Peres, Ann. Probab. 2013], this has applications to diffusive estimates for random walks on …
Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise
考虑了在d维时空白噪声驱动下的空间维1的d非线性随机热方程组。非线性既表现为加性漂移项，又表现为噪声的乘数。利用Malliavin演算的技巧，我们建立了解u（t，x）的单点密度的上界和下界，以及（u（s，y），u（t，x））的两点密度的高斯型上界.特别地，该估计量化了该密度如何随着（s，y）收敛到（t，x）而退化。由这些结果，我们分别根据Hausdorff测度和牛顿容量，导出了过程{u（t，x）}_{tPDMATHABMM \mathbb{R}_+，x \in [0，1]}的命中概率的上、下界.这些估计使得证明当d> 6时点是极坐标，而当d 6时点不是极坐标成为可能，并对过程t \mapsto u（t，x）和x \mapsto u（t，x）给出类似的结果.<6. We also show that the Hausdorff dimension of the range of the process is 6 when d>最后，我们得到了这些过程的水平集的豪斯多夫维数值.
We study bond percolation for a family of infinite hyperbolic graphs. We relate percolation to the appearance of homology in finite versions of these graphs. As a consequence, we derive an upper bound on the critical probabilities of the infinite graphs.
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…
The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including speed of mixing, tails of decay of the connectivity probabilities, scaling relations, etc. In this article, we develop a renormalization scheme for crossing probabilities in the two-dimensional random-cluster model. The outcome of the process is a precise description of an alternative between four behaviors: - Subcritical: Crossing probabi…
We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known that the Azuma-Hoeffding inequality holds, which gives an upper bound of a probability for exceptional events. The inequality gives a simple form of the upper bound, and it has been utilized for many investigations. However, the inequality is not tight. We give an explicit expression of a tight upper bound, and we show that it and the bound obtained from th…
We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of [DCHN11] and [CS12]. Our result relies on new discrete complex analysis techniques, introduced in [Che12]. We detail some applications, in particular the computation of so-called universal exponents, the proof of quasi-multiplicativity properties of arm probabilities, and bounds on crossing probabilities for the classical Ising model.
