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.

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.