Summary
A special (“extended”) kind of convergence in distribution of processes with filtration is considered. Recent theorems on the functional convergence of semimartingales are improved by showing that their assumptions imply the extended convergence of semimartingales to continuous in probability processes with independent increments.
References
Aldous, D.J.: Stopping times and tightness. Ann. Probab. 6, 335–340 (1978)
Aldous, D.J.: A concept of weak convergence for stochastic processes viewed in the Strasbourg manner. Preprint, Statist. Laboratory, Univ. Cambridge (1979)
Bilingsley, P.: Weak Convergence of Probability Measures. New York: Wiley 1968
Grigelionis, B.: On the martingale characterization of stochastic processes with independent increments. Lietuwos Matematikos Rinkinys XVII, N0 1, 75–86 (1977)
Helland, I.S.: On Weak Convergence to Brownian Motion. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 52, 251–265 (1980)
Ikeda, N., Watanabe, S.: Stochastic Differential Equation and Diffusion Processes. Amsterdam-Oxford-New York: North-Holland Publishing Company 1981
Jacod, J.: Calcul stochastique et problèmes de martingales. Lecture Notes in Math., N0 714. Berlin-Heidelberg-New York: Springer 1979
Jacod, J.: Processus à accroissements indépendants: une condition nécessaire et suffisante de convergence en loi. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 63, 109–136 (1983)
Jacod, J., Kłopotowski, A., Memin, J.: Théorème de la limite centrale et convergence fonctionnelle vers un processus à accroissements indépendants: la méthode des martingales. Ann. I.H.P., XVIII, N0 1, 1–45 (1982)
Jakubowski, A.: Limit theorems for sums of dependent random variables with values in a Hilbert space. Thesis, (in Polish) Toruń, (1982)
Lindvall, T.: Weak convergence of probability measures and random functions in the function space D [0, ∞]. J. Appl. Probab. 10, 109–121 (1973)
Liptser, R., Shiryaev, A.: Necessary and sufficient conditions for the functional central limit theorem for semimartingales. Theory Probab. Appl. 26, 132–137 (1981)
Liptser, R., Shiryaev, A.: On a Problem of Necessary and Sufficient Conditions in the Functional Central Limit Theorem for Local Martingales. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 59, 311–318 (1982)
Słomiński, L.: Necessary and sufficient conditions for extended convergence of semimartingales. Preprint (1984)
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Jakubowski, A., Słomiński, L. Extended convergence to continuous in probability processes with independent increments. Probab. Th. Rel. Fields 72, 55–82 (1986). https://doi.org/10.1007/BF00343896
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DOI: https://doi.org/10.1007/BF00343896
Keywords
- Filtration
- Stochastic Process
- Probability Theory
- Statistical Theory
- Probability Process