Limit Theorems for Randomly Stopped Stochastic Processes

ISBN-13: 9781447110514 / Angielski / Miękka / 2012 / 398 str.

Dmitrii S. Silvestrov; Dmitrii S. Silvestrov

Opis

This volume is the first to present a state-of-the-art overview of this field, with many results published for the first time. It covers the general conditions as well as the basic applications of the theory, and it covers and demystifies the vast and technically demanding Russian literature in detail. Its coverage is thorough, streamlined and arranged according to difficulty.

Spis treści

1 Weak convergence of stochastic processes.- 1.1 Introductory remarks.- 1.2 Weak convergence in Rm.- 1.3 Weak convergence in metric spaces.- 1.4 The space D of càdlàg functions.- 1.5 J-continuous functionals.- 1.6 J-convergence of càdlàg processes.- 2 Weak convergence of randomly stopped stochastic processes.- 2.1 Introductory remarks.- 2.2 Randomly stopped scalar càdlàg processes.- 2.3 Randomly stopped vector càdlàg processes.- 2.4 Weakened continuity conditions.- 2.5 Iterated weak limits.- 2.6 Scalar compositions of càdlàg processes.- 2.7 Vector compositions of càdlàg processes.- 2.8 Translation theorems.- 2.9 Randomly stopped locally compact càdlàg processes.- 3 J-convergence of compositions of stochastic processes.- 3.1 Introductory remarks.- 3.2 Compositions with asymptotically continuous components.- 3.3 Asymptotically continuous external processes.- 3.4 Asymptotically continuous internal stopping processes.- 3.5 Semi-vector compositions of càdlàg functions.- 3.6 Semi-vector compositions of càdlàg processes.- 3.7 Vector compositions of càdlàg functions.- 3.8 Vector compositions of càdlàg processes.- 4 Summary of applications.- 4.1 Introductory remarks.- 4.2 Randomly stopped sum-processes.- 4.3 Generalised exceeding processes.- 4.4 Step generalised exceeding processes.- 4.5 Sum-processes with renewal stopping.- 4.6 Accumulation processes.- 4.7 Extremes with random sample size.- 4.8 Mixed sum-max processes.- 4.9 Max-processes with renewal stopping.- 4.10 Shock processes.- Bibliographical remarks.- References.

O autorze

Limit theorems for stochastic processes are an important part of probability theory and mathematical statistics and one model that has attracted the attention of many researchers working in the area is that of limit theorems for randomly stopped stochastic processes.
This volume is the first to present a state-of-the-art overview of this field, with many of the results published for the first time. It covers the general conditions and theorems as well as the basic applications of the theory, and it covers and demystifies the vast, and technically demanding, Russian literature in detail. A survey of the literature and an extended bibliography of works in the area are also provided.
The coverage is thorough, streamlined and arranged according to difficulty so that the book can be used as an upper-level text. It is an essential reference for theoretical and applied researchers in the fields of probability and statistics that will contribute to the continuing extensive studies in the area and remain relevant for years to come.