Small Stochastic Perturbations of Random Maps with Position Dependent Probabilities

Abstract

A position dependent random map is a dynamical system consisting of a collection of maps such that, at each iteration, a selection of a map is made randomly by means of probabilities which are functions of position. Let f* be an invariant density of the position dependent random map T. We consider a model of small random perturbations 𝔗_? of the random map T. For each ? > 0, 𝔗_? has an invariant density function f ^?. We prove that f ^? → f* as ? → 0.     

Absolutely continuous invariant measures for random maps with position dependent probabilities

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τ_k is chosen from a finite collection of maps with constant probability p_k. In this note we allow the p_k's to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0,1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.     

Attainable densities for random maps

We consider a random map T=T(Γ,ω), where Γ=(τ₁,τ₂,…,τK) is a collection of maps of an interval and ω=(p₁,p₂,…,pK) is a collection of the corresponding position dependent probabilities, that is, pk(x)?0 for k=1,2,…,K and . At each step, the random map T moves the point x to τk(x) with probability pk(x). For a fixed collection of maps Γ, T can have many different invariant probability density functions, depending on the choice of the (weighting) probabilities ω. Most of the results in this paper concern random maps where Γ is a family of piecewise linear semi-Markov maps. We investigate properties of the set of invariant probability density functions of T that are attainable by allowing the probabilities in ω to vary in a certain class of functions. We prove that the set of all attainable densities can be determined algorithmically. We also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches. Such representation may be of greater interest in view of new methods of computing entropy [W. S?omczyński, J. Kwapień, K. ?yczkowski, Entropy computing via integration over fractal measures, Chaos 10 (2000) 180-188].     

Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

In this paper, we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large-deviation prob- abilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang （J. Appl. Prob., 41, 93-107, 2004）.     

Statistical maps and generalized random walks

We continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.     

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

ABSTRACT

In this paper, for centred homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands' grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.     

Lyapunov Exponents for Position Dependent Random Maps: Formulae and Applications

In this article, we provide formulae for Lyapunov exponents of position dependent random maps of the interval. We then apply our results to a financial market model with short-lived assets.     

Hoeffding’s Inequality for Sums of Dependent Random Variables

Let \(X_1,\ldots ,X_n\) be, possibly dependent, [0, 1]-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper, we provide a generalisation of Hoeffding’s theorem. We obtain an estimate on the aforementioned probability that is described in terms of the expectation, with respect to convex functions, of a random variable that concentrates mass on the set \(\{0,1,\ldots ,n\}\). Our main result yields concentration inequalities for several sums of dependent random variables such as sums of martingale difference sequences, sums of k-wise independent random variables, as well as for sums of arbitrary [0, 1]-valued random variables.     

Boundaries of random walks on graphs and groups with infinitely many ends

Consider an irreducible random walk {Z _n} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Z _n} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Z _n} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Z _n} converges to this end. If {Z _n} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.     

Performance Analysis of a Finite-Buffer Bulk-Arrival and Bulk-Service Queue with Variable Server Capacity

Abstract

In this paper, we consider a finite-buffer bulk-arrival and bulk-service queue with variable server capacity: M ^X /G ^Y /1/K + B. The main purpose of this paper is to discuss the analytic and computational aspects of this system. We first derive steady-state departure-epoch probabilities based on the embedded Markov chain method. Next, we demonstrate two numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as moments of the number of customers in the queue at three different epochs, the loss probability, and the probability that server is busy. Numerical results are presented for a deterministic service-time distribution – a case that has gained importance in recent years.     

Almost Sure Convergence for Linear Processes Generated by Positively Dependent Processes

Abstract

In this article we obtain strong laws of large numbers for linear processes generated by sequences of random variables, which are either linearly positive quadrant dependent or associated. No stationarity is required.     

Uniform asymptotics for the finite-time and infinite-time ruin probabilities in a dependent risk model with constant interest rate and heavy-tailed claims

We consider a nonstandard risk model with constant interest rate. For the case where the claim sizes follow a common heavy-tailed distribution and fulfill a dependence structure proposed by Geluk and Tang [J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theor. Probab., 22:871–882, 2009] while the interarrival times fulfill the so-called widely lower orthant dependence, we establish a weakly asymptotically equivalent formula for the infinite-time ruin probability. In particular, when the dependence structure for claim sizes is strengthened to the widely upper orthant dependence, this result implies a uniformly asymptotically equivalent formula for the finite-time and infinite-time ruin probabilities.     

时间相依更新风险模型中无限时绝对破产概率的渐近性

本文考虑了两类时间相依且带常利率和常值保费收入率的更新风险模型的无限时绝对破产概率, 其中索赔额及其到达时间间隔构成独立同分布的随机对列, 以及每个随机对遵循某种相依结构. 基于此, 当索赔额分布属于R_-∞∩J(γ), γ > 0 分布族时, 我们分别得到了两类时间相依结构下的无限时绝对破产概率的渐近公式和渐近上界.     

Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract

We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].     

Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

Note on “Obtaining the Maximum Likelihood Estimates in Incomplete R × C Contingency Tables Using a Poisson Generalized Linear Model”

Abstract

This note extends the construction of the design matrix used for estimating cell probabilities with ignorable missing data described by Lipsitz, Parzen, and Molenberghs. A reformulation for the general case of an n-way table is described and implemented in a SAS macro program. The macro constructs this design matrix and offset variable, estimates the cell probabilities, and returns a table with the estimates, their standard errors, and fitted cell frequencies.     

Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d ≥ 3. The method of proof is by martin‐gale techniques which rely on the fact that symmetric random walks are transient in high dimensions. © 2000 John Wiley & Sons, Inc.     

Random approximations and random fixed point theorems for random 1-setn-contractive non-self-maps in abstract cones

In this paper, we will prove that the random version of Fan's Theorem [6, Theorem 2] is true for a random hemicompact 1-set-contractive map defined on a closed ball, a sphere and an annulus in cones. This class of random 1-set-contractive map includes random condensing maps, random continuous semicontractive maps, random LANE maps, random nonexpansive maps and others. As applications of our theorems, some random fixed point theorems of non-self-maps are proved under various well-known boundary conditions. Our results are generalizations, improvements or stochastic versions of the recent results obtained by many authors     

A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments

Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B 47, 125–145, but under the assumption of second moments only. This allows us to derive Lindeberg–Feller type theorems for sums of random vectors with certain dependence structures. We apply the main theorem to the study of three problems: local dependence, random graph degree statistics and finite population statistics. In particular, we consider U-statistics of independent observations as well as of observations drawn without replacement.     

Inventory System Subject to Disaster with General Interarrival Times

Abstract

We discuss a single commodity continuous review (s, S) inventory system in which commodities get damaged due to external disaster. Shortages are not permitted and lead time is assumed to be zero. The interarrival times of demands constitute a family of i.i.d. random variables with a common arbitrary distribution. The quantity demanded at a demand epoch is arbitrarily distributed which depends only on the time elapsed since the last demand epoch. Transient and steady state probabilities of the inventory levels are derived by identifying suitable semi-regenerative process. In the case when the demand is for unit item and the disaster affects only an exhibiting item, the steady state probability distribution is obtained as uniform. An optimization problem is discussed and numerical examples are provided.