Abstract:
The first part of the paper contains a urvey of tne present state of research in the general problems of convergence of stochastic processes. An important place is given to theorems on weak convergence of distributions on metric spaces. Then a ifferent approach is proposed to the study of convergence of distributions of functionals on stochastic processes, which is connected with the approximation of the trajectories of a rocess by some family of functions. We think of approximation in terms of the nearness of the functionals in question. Using this approach we obtain all the main results on convergence in specific function spaces that are known at present. These results are obtained in their most general form without the requirement that the limiting processes should belong to the space under discussion. New limit theorems are also obtained and among them theorems for processes with discontinuities of the second kind and others.
\Bibitem{Bor72}
\by A.~A.~Borovkov
\paper The convergence of tributions of functionals on stochastic processes
\jour Russian Math. Surveys
\yr 1972
\vol 27
\issue 1
\pages 1--42
\mathnet{http://mi.mathnet.ru/eng/rm5005}
\crossref{https://doi.org/10.1070/RM1972v027n01ABEH001362}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=400325}
\zmath{https://zbmath.org/?q=an:0255.60005}
Linking options:
https://www.mathnet.ru/eng/rm5005
https://doi.org/10.1070/RM1972v027n01ABEH001362
https://www.mathnet.ru/eng/rm/v27/i1/p3
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A. A. Mogul'skiǐ, “The Extended Large Deviation Principle for the Trajectories of a Compound Renewal Process”, Sib. Adv. Math., 32:1 (2022), 35
A. A. Mogulskii, “Rasshirennyi printsip bolshikh uklonenii dlya traektorii obobschennogo protsessa vosstanovleniya”, Matem. tr., 24:1 (2021), 142–174
F. C. Klebaner, A. V. Logachov, A. A. Mogulskii, “Extended large deviation principle for trajectories of processes with independent and stationary increments on the half-line”, Problems Inform. Transmission, 56:1 (2020), 56–72
A. A. Borovkov, “Functional limit theorems for compound renewal processes”, Siberian Math. J., 60:1 (2019), 27–40
F. C. Klebaner, A. A. Mogulskii, “Large deviations for processes on half-line: Random Walk and Compound Poisson Process”, Sib. elektron. matem. izv., 16 (2019), 1–20
A. A. Mogul'skiǐ, “The extended large deviation principle for a process with independent increments”, Siberian Math. J., 58:3 (2017), 515–524
A. A. Mogul'skiǐ, “The large deviation principle for a compound Poisson process”, Siberian Adv. Math., 27:3 (2017), 160–186
A. A. Mogul'skiǐ, “The expansion theorem for the deviation integral”, Siberian Adv. Math., 23:4 (2013), 250–262
A. A. Borovkov, A. A. Mogul'skii, “On large deviation principles for random walk trajectories. II”, Theory Probab. Appl., 57:1 (2013), 1–27
A. A. Borovkov, “Large deviation principles for random walks with regularly varying distributions of jumps”, Siberian Math. J., 52:3 (2011), 402–410
A. A. Borovkov, A. A. Mogul'skiǐ, “Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks”, Siberian Math. J., 52:4 (2011), 612–627
D. S. Silvestrov, “Convergence in skorokhod $J$-topology for
compositions of stochastic processes”, Theory Stoch. Process., 14(30):1 (2008), 126–143
Heinz Cremers, Dieter Kadelka, “On weak convergence of stocastic processes with Lusin path spaces”, manuscripta math, 45:2 (1984), 115
A. A. Borovkov, “Convergence of measures and random processes”, Russian Math. Surveys, 31:2 (1976), 1–69
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