Extended renovation theory and limit theorems for stochastic ordered graphs∗ SERGUEI FOSS TAKIS KONSTANTOPOULOS
ثبت نشده
چکیده
We extend Borovkov’s renovation theory to obtain criteria for coupling-convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an “infinite bin model”, a particular system that is an abstraction of a stochastic ordered graph, i.e., a graph on the integers that has (i, j), i < j, as an edge, with probability p, independently from edge to edge. A question of interest is an estimate of the length Ln of a longest path between two vertices at distance n. We give sharp bounds on C = limn→∞(Ln/n). This is done by first constructing the unique stationary version of the infinite bin model, using extended renovation theory. We also prove a functional law of large numbers and a functional central limit theorem for the infinite bin model. Finally, we discuss perfect simulation, in connection to extended renovation theory, and as a means for simulating the particular stochastic models considered in this paper.
منابع مشابه
Extended Renovation Theory and Limit Theorems for Stochastic Ordered Graphs ∗
We extend Borovkov’s renovation theory to obtain criteria for coupling-convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an “infinite bin model”, a particular system that is an abstraction of a stochastic ordered graph, i.e., a graph on the integers that has (i, j), i < j, as an edge, with probability p, independently from edge to...
متن کاملLimit theorems for a random directed slab graph
We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability pj−i depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a specia...
متن کاملAn Overview of Some Stochastic Stability Methods ̃
This paper presents an overview of stochastic stability methods, mostly motivated by (but not limited to) stochastic network applications. We work with stochastic recursive sequences, and, in particular, Markov chains in a general Polish state space. We discuss, and frequently compare, methods based on (i) Lyapunov functions, (ii) fluid limits, (iii) explicit coupling (renovating events and Har...
متن کاملThe principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments
We consider a modulated process S which, conditional on a background process X , has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in cont...
متن کامل