Geodesic
Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1091-1098 
A. N. Nazarova. Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1091-1098. http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a10/
@article{FPM_2002_8_4_a10,
     author = {A. N. Nazarova},
     title = {Logarithmic velocity of convergence in {CLT} for stochastic linear processes and fields in {a~Hilbert} space},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1091--1098},
     year = {2002},
     volume = {8},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a10/}
}
TY  - JOUR
AU  - A. N. Nazarova
TI  - Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2002
SP  - 1091
EP  - 1098
VL  - 8
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a10/
LA  - ru
ID  - FPM_2002_8_4_a10
ER  - 
%0 Journal Article
%A A. N. Nazarova
%T Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2002
%P 1091-1098
%V 8
%N 4
%U http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a10/
%G ru
%F FPM_2002_8_4_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper the sums of linear random fields defined on $\mathbb Z^d$ for any $d\geqslant1$ and taking values in a Hilbert space are studied. The convergence velocity in CLT for such fields is discussed. We obtain easily verifiable sufficient conditions for logarithmic velocity of convergence.

[1] Bentkus V., “Asimptoticheskie razlozheniya dlya raspredelenii summ nezavisimykh elementov prostranstva Gilberta”, Litovskii mat. sbornik, 24:4 (1984), 29–48 | MR | Zbl

[2] Bentkus V., Gettse F., Paulauskas V., Rachkauskas A., “Tochnost gaussovskoi approksimatsii v banakhovykh prostranstvakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 81, VINITI, M., 1991, 39–139 | MR

[3] Zalesskii B. A., “O skorosti skhodimosti v tsentralnoi predelnoi teoreme na nekotorom klasse mnozhestv gilbertova prostranstva”, Teor. ver. i ee primeneniya, 30:4 (1985), 662–670 | MR | Zbl

[4] Zalesskii B. A., Sazonov V. V., Ulyanov V. V., “Pravilnaya otsenka skorosti skhodimosti v tsentralnoi predelnoi teoreme v gilbertovom prostranstve”, Mat. sbornik, 180:12 (1989), 1587–1613 | Zbl

[5] Zolotarev V. M., “Absolyutnaya otsenka ostatochnogo chlena v tsentralnoi predelnoi teoreme”, Teor. ver. i ee primeneniya, 11:1 (1966), 108–119 | MR | Zbl

[6] Yurinskii V. V., “O tochnosti normalnogo priblizheniya veroyatnosti popadaniya v shar”, Teor. ver. i ee primeneniya, 27:2 (1982), 270–278 | MR | Zbl

[7] Senatov V. V., Normal approximation: new results, methods and problems, Modern Probability and Statistics, VSP, Utrecht, 1999 | MR

[8] Nazarova A. N., “Normalnaya approksimatsiya dlya sluchainykh lineinykh protsessov i polei v gilbertovom prostranstve”, Mat. zametki, 68:3 (2000), 421–428 | MR | Zbl

[9] Araujo A., Giné E., The central limit theorem for real and Banach valued random variables, John Wiley and Sons, New York, 1980 | MR | Zbl

[10] Kuelbs J., Kurtz T., “Berry–Essen estimates in Hilbert space and applications to the law of the iterated logarithm”, Ann. Probability, 2:3 (1974), 387–407 | DOI | MR | Zbl