Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $ν_n = \sum_i ξ_i $, where $ξ_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $ξ_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a law of large numbers and central limit theorem for $ν_n(f)$. The latter implies weak convergence of $ν_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications including the volume and surface measure of germ-grain models with unbounded grain sizes.
On the convergence, lock-in probability and sample complexity of stochastic approximation
证明了在标准假设下，如果随机逼近迭代保持紧，则它们以概率1收敛到它们的o. d. e。极限提示一个简单的密封性测试（因此，收敛）。此外，对锁定概率的估计，即，收敛到O.D.E.的特定吸引子的概率限制给定的迭代访问其吸引区域和样本复杂性，即，还提供了在具有规定概率的期望极限集的规定邻域内所需的步数。后者改进显着现有的结果，因为他们需要一个更弱的条件，鞅差噪声。
This paper considers an approximation usually used when implementing Ramaswami's recursion for the stationary distribution of the M/G/1-type Markov chain. The approximation is called the level-increment-truncation approximation because it truncates level increment at a given threshold. The main contribution of this paper is to present a geometric convergence formula of the level-wise difference between the respective stationary distributions of the original M/G/1-type Markov chain and its LI truncation approximation under the assumption that the level-increment distribution is light-tailed.
Let $(X_n)_{n=0}^\infty$ denote a Markov chain on a Polish space that has a stationary distribution $\varpi$. This article concerns upper bounds on the Wasserstein distance between the distribution of $X_n$ and $\varpi$. In particular, an explicit geometric bound on the distance to stationarity is derived using generalized drift and contraction conditions whose parameters vary across the state space. These new types of drift and contraction allow for sharper convergence bounds than the standard versions, whose parameters are constant. Application of the result is illustrated in the context of …
[Błaszczyszyn, Yogeshwaran and Yukich (2019)] established central limit theorems for geometric statistics of point processes having fast decay dependence. As limit theorems are of limited use unless we understand their errors involved in the approximation, in this paper, we consider the rates of a normal approximation in terms of the Wasserstein distance for statistics of point processes on $\mathbb{R}^d$ satisfying fast decay dependence. We demonstrate the use of the theorems for statistics arising from two families of point processes: the rarified Gibbs point processes and the determinantal …
We introduce and study deferred Nörlund statistical convergence in probability, mean of order $r,$ distribution, and study the interrelation among them. Based upon the proposed method to illustrate the findings, we present new Korovkin type theorems for the sequence of random variables via deferred Nörlund statistically convergence and present compelling examples to demonstrate the effectiveness of the results.
Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor
描述由某种递推公式控制的离散时间随机过程的渐近行为的一个成功方法是将它与一个精心选择的平均微分方程的极限集联系起来。在可达性条件下，证明了由该动力系统引起的流的收敛到给定吸引子的概率为正（Benaïm，1999）。Benaïm et al.（2005）将这种方法推广到随机逼近算法，其平均行为与微分包含有关。我们追求的类比延伸到这个设置的结果与正概率收敛到吸引子。
We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation a…
While the Metropolis Adjusted Langevin Algorithm (MALA) is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors' knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish $V$-uniformly geometric convergence for MALA under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quit…
Consider $n$ points independently sampled from a density $p$ of class $\mathcal{C}^2$ on a smooth compact $d$-dimensional sub-manifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the generator of a random walk visiting these points according to a transition kernel $K$. We study the almost sure uniform convergence of this operator to the diffusive Laplace-Beltrami operator when $n$ tends to infinity. This work extends known results of the past 15 years. In particular, our result does not require the kernel $K$ to be continuous, which covers the cases of walks exploring $k$NN-random and geomet…
