It is well-known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also give explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.
Stochastically monotone Markov chains arise in many applied domains, especially in the setting of queues and storage systems. Poisson's equation is a key tool for analyzing additive functionals of such models, such as cumulative sums of waiting times or sums of rewards. In this paper, we show that when the reward function for such a Markov chain is monotone, the solution of Poisson's equation is monotone. This implies that the value function associated with infinite horizon average reward is monotone in the state when the reward is monotone.
Semilinear stochastic evolution equations with multiplicative Poisson noise and monotone nonlinear drift are considered. We do not impose coercivity conditions on coefficients. A novel method of proof for establishing existence and uniqueness of the mild solution is proposed. Examples on stochastic partial differential equations and stochastic delay differential equations are provided to demonstrate the theory developed.
Semilinear stochastic evolution equations with Lévy noise and monotone nonlinear drift are considered. The existence and uniqueness of the mild solutions in $L^p$ for these equations is proved and a sufficient condition for exponential asymptotic stability of the solutions is derived. The main tool in our study is an Itô type inequality for the $p$th power of stochastic convolution integrals in Hilbert spaces.
In [8] we established existence and uniqueness of solutions of backward stochastic differential equations in L^p under a monotonicity condition on the generator and in a general filtration. There was a mistake in the case 1 \textless{} p \textless{} 2. Here we give a corrected proof. Moreover the quasi-left continuity condition on the filtration is removed.
We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions in $L^p$ provided that the generator and the terminal condition satisfy appropriate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principl…
In this note we consider the Glauber dynamics for the mean-field Ising model, when all couplings are equal and the external field is uniform. It is proved that the relaxation time of the dynamics is monotonically decreasing in temperature.
We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the harmonic approximation result from \cite{GHO} with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments.
We consider a class of probability measures $μ_{s,r}^α$ which have explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $μ_{s,2}^α$ as a free compound Poisson law with Lévy measure a monotone $α$-stable law. This implies the free infinite divisibility of $μ_{s,2}^α$. Moreover, when symmetric or positive, $μ_{s,2}^α$ has a representation as the free multiplication of a free Poisson law and a monotone $α$-stable law. We also investigate the free infinite divisibility of $μ_{s,r}…
We study monotone and convex stochastic orders for processes with independent increments. Our contributions are twofold: First, we relate stochastic orders of the Poisson component to orders of their (generalized) Lévy measures. The relation is proven using an interpolation formula for infinitely divisible laws. Second, we derive explicit conditions on the characteristics of the processes. In this case, we prove the conditions via constructions of couplings.
The asymptotic analysis of a class of stochastic partial differential equations (SPDEs) with fully locally monotone coefficients covering a large variety of physical systems, a wide class of quasilinear SPDEs and a good number of fluid dynamic models is carried out in this work. The aim of this work is to develop the large deviation theory for small Gaussian as well as Poisson noise perturbations of the above class of SPDEs. We establish a Wentzell-Freidlin type large deviation principle for the strong solutions to such SPDEs perturbed by Lévy noise in a suitable Polish space using a variation…
We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,π]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random variables. In this note, we provide estimates for the probability that the sum of independent monotone integer variables attains precisely a specific value. We do not assume that the variables are identically distributed. Our estimates are sharp when the specific value is close to the mean, but they are not useful further out in the tail. By combining with the …
In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of $\mathbb{L}^p(p\ge2)$-solutions in the case of a fixed terminal time under suitable $p$-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension $1$.
We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition $P$ we assign a weight depending on the numbers of disorders and orders in $P$ related to the natural partial order on the set of blocks of $P$ implemented by the relation of being inner or outer. In particular, we obtain a simp…
Attenuated Poisson Dirichlet approximations for divisibility configurations
Centro de Investigación en matemáticas (CIMAT)（数学研究中心）；Escuela Nacional de Estudios Superiores, Unidad León, Guanajuato（全国高等研究学校莱昂分校）；UC Santa Barbara（加州大学圣塔芭芭拉分校）
Given a sequence of $n$ independent random variables with common continuous distribution, we propose a simple adaptive online policy that selects a monotone increasing subsequence. We show that the expected number of monotone increasing selections made by such a policy is within $O(\log n)$ of optimal. Our construction provides a direct and natural way for proving the $O(\log n)$-optimality gap. An earlier proof of the same result made crucial use of a key inequality of Bruss and Delbaen (2001) and of de-Poissonization.
Consider a sequence of $n$ independent random variables with a common continuous distribution $F$, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy $π_n^*$ that is optimal in the sense that it maximizes the expected value of $L_n(π_n^*)$, the number of selected observations. We investigate the distribution of $L_n(π_n^*)$; in particular, we obtain a central limit theorem for $L_n(π_n^*)$ and a detailed understanding of i…
We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order $α(t)$ varying with time $t\geq0$. In particular, we use the operator introduced by Scarpi in the Seventies and later reformulated in the regularized Caputo sense in Garrappa et al. (2021), inside the framework of the so-called general fractional calculus. The model obtained extends the well-known time-fractional Poisson process of fixed order $α\in (0,1)$ and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memo…
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time-homogeneous with $σ$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.
