Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables

Albeverio, Sergio; Liang, Song

Ann. Probab., Tome 33 (2005) no. 1, p. 300-336 / Harvested from Project Euclid

Résumé

Let X_i, i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a smooth enough mapping from B into R. An asymptotic evaluation of Z_n=E(exp(nΦ(∑_i=1ⁿX_i/n))), up to a factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields 72 (1986) 305–318] and Kusuoka and Liang [Probab. Theory Related Fields 116 (2000) 221–238]. In this paper, a detailed asymptotic expansion of Z_n as n→∞ is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.

Publié le : 2005-01-14
Classification: Laplace approximation, asymptotic expansions, i.i.d. random vectors, Banach space-valued random variables, 62E20, 60F10, 60B12

@article{1108141728,
     author = {Albeverio, Sergio and Liang, Song},
     title = {Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables},
     journal = {Ann. Probab.},
     volume = {33},
     number = {1},
     year = {2005},
     pages = { 300-336},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1108141728}
}

Albeverio, Sergio; Liang, Song. Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables. Ann. Probab., Tome 33 (2005) no. 1, pp.  300-336. http://gdmltest.u-ga.fr/item/1108141728/