Geodesic
Weak convergence of Hilbert-valued semimartingales to a stochastically continuous process with independent increments
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2024), pp. 5-16 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper studies the weak convergence of semimartingales taking values in Hilbert space to an arbitrary stochastically continuous process with independent increments. Sufficient conditions for the weak convergence of such semimartingales to a stochastically continuous semimartingale with independent increments are obtained.
Keywords: semimartingale, Hilbert space, weak convergence, stochastically continuous processes. 
@article{VTPMK_2024_1_a0,
     author = {V. V. Lavrentyev},
     title = {Weak convergence of {Hilbert-valued} semimartingales to a stochastically continuous process with independent increments},
     journal = {Vestnik Tverskogo gosudarstvennogo universiteta. Seri\^a Prikladna\^a matematika},
     pages = {5--16},
     year = {2024},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTPMK_2024_1_a0/}
}
TY  - JOUR
AU  - V. V. Lavrentyev
TI  - Weak convergence of Hilbert-valued semimartingales to a stochastically continuous process with independent increments
JO  - Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika
PY  - 2024
SP  - 5
EP  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VTPMK_2024_1_a0/
LA  - ru
ID  - VTPMK_2024_1_a0
ER  - 
%0 Journal Article
%A V. V. Lavrentyev
%T Weak convergence of Hilbert-valued semimartingales to a stochastically continuous process with independent increments
%J Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika
%D 2024
%P 5-16
%N 1
%U http://geodesic.mathdoc.fr/item/VTPMK_2024_1_a0/
%G ru
%F VTPMK_2024_1_a0
V. V. Lavrentyev. Weak convergence of Hilbert-valued semimartingales to a stochastically continuous process with independent increments. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2024), pp. 5-16. http://geodesic.mathdoc.fr/item/VTPMK_2024_1_a0/

[1] Liptser R. Sh., Shiryaev A. N., “On weak convergence of semimartingales to stochastically continuous processes with independent and conditionally independent increments”, Mathematics of the USSR-Sbornik, 44:3 (1983), 299–323 | DOI | MR | Zbl | Zbl

[2] Liptser R. Sh., Shiryaev A. N., Theory of martingales, Mathematics and its Applications (Soviet Series), v. 49, Kluwer Academic Publishers Group, Dordrecht, 1989, 792 pp. | MR | MR

[3] Prokhorov Yu. V., “Convergence of Random Processes and Limit Theorems in Probability Theory”, Theory of Probability and its Applications, v. 1, Torus Press, Moscow, 1956, 157–214 | DOI | MR

[4] Lavrentyev V. V., Nazarov L. V., “A functional central limit theorem for Hilbert-valued martingales”, Lobachevskii Journal of Mathematics, 37 (2016), 138–145 | DOI | MR | Zbl

[5] Lavrentyev V. V., Nazarov L. V., “Weak convergence to Gaussian martingale of semimartingales with values in Hilbert space”, Herald of Tver State University. Series: Applied Mathematics, 2015, no. 3, 45–57 (in Russian)

[6] Lavrentyev V. V., “On the structure of Hilbert-valued martingales”, Herald of Tver State University. Series: Applied Mathematics, 2010, no. 2, 13–19 (in Russian)

[7] Lavrentyev V. V., “Canonical representation of Hilbert-valued semi-martingales”, Herald of Tver State University. Series: Applied Mathematics, 2011, no. 1, 123–130 (in Russian)

[8] Meyer P. A., “Notes sur les integrales stochastiques. I Integrales Hilbertiennes”, Lecture Notes in Mathematics, 581 (1977), 446–462 | DOI | MR | Zbl

[9] Lavrentyev V. V., Bugrimov A. L., “Compactness conditions for a family of measures of Hilbert-valued continuous semi-martingales”, Herald of Tver State University. Series: Applied Mathematics, 2019, no. 4, 39–51 (in Russian)

[10] Jacod J., Mémin J., “Sur la convergence des semimartingales vers un processus à accroissements indépendants”, Lecture Notes in Mathematics, 784 (1980), 227–248 | DOI | MR | Zbl

[11] Billingsli P., Convergence of probability measures, Nauka Publ., Moscow, 1977, 352 pp. (in Russian) | MR

[12] Varadhan S. R. S., “Limit theorems for sums of independent random variables with values in a Hilbert space”, Sankhyā: The Indian Journal of Statistics, Series A, 24:3 (1962), 213–238 | MR | Zbl

[13] Jajte R., “On convergence of infinitely divisiable distributions in a Hilbert space”, Colloquium Mathematicum, 19:2 (1968), 327–332 | DOI | MR | Zbl