Replication and Its Application to Weak Convergence

2020/11/01 by Chi Dong, Dong, Chi, Michael A. Kouritzin +1
Computer Science · Mathematics · #60B05 #60B10 #60F17 #60G57 #Cellular Automata and Applications #FOS: Mathematics #Primary 60G07 Secondary 60A10 #Probability (math.PR) #math.PR #msc:60A10 #msc:60B05 #msc:60B10 #msc:60F17 #msc:60G07 #msc:60G57

paper · pdf · doi:10.48550/arxiv.2011.00484

203 pages, 5 figures

arxiv created 2020/11/01 · openalex publication_date 2020/11/01 · arxiv updated 2020/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Herein, a methodology is developed to replicate functions, measures and stochastic processes onto a compact metric space. Many results are easily established for the replica objects and then transferred back to the original ones. Two problems are solved within to demonstrate the method: (1) Finite-dimensional convergence for processes living on general topological spaces. (2) New tightness and relative compactness criteria are given for the Skorokhod space D(R⁺;E) with E being a general Tychonoff space. The methods herein are also used in companion papers to establish the: (3) existence of, uniqueness of and convergence to martingale problem solutions, (4) classical Fujisaki-Kallianpur-Kunita and Duncan-Mortensen-Zakai filtering equations and stationary filters, (5) finite-dimensional convergence to stationary signal-filter pairs, (6) invariant measures of Markov processes, and (7) Ray-Knight theory all in general settings.

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