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 タイトル: Laplace Approximations for Sums of Independent Random Vectors -- The Degenerate Case -- 著者: Liang, Song 発行日: 2000年 出版者: Graduate School of Mathematical Sciences, The University of Tokyo 掲載誌情報: Journal of Mathematical Sciences, The University of Tokyo. Vol. 7 (2000), No. 2, Page 195-220 抄録: Let $X_i, i \in {\bf N}$, be {\it i.i.d.} $B$-valued random variables, where $B$ is a real separable Banach space. Let $Φ: B \to {\bf R}$ be a mapping. The problem is to give an asymptotic evaluation of $Z_n = E \left( \exp \left( n Φ (\sum_{i=1}^n X_i / n ) \right) \right)$, up to a factor $(1 + o(1))$. Bolthausen \cite{Bolthausen} studied this problem in the case that there is a unique point maximizing $Φ - h$, where $h$ is the so-called entropy function, and the curvature at the maximum is nonvanishing, (these two will be called as {\it nondegenerate assumptions}), with some central limit theorem assumption. Kusuoka-Liang \cite{K-L} studied the same problem, and succeeded in eliminating the central limit theorem assumption, but the nondegenerate assumptions are still left. In this paper, we study the same problem not assuming the central limit theorem assumption and the nondegenerate assumptions. URI: http://hdl.handle.net/2261/1214 ISSN: 13405705 出現カテゴリ: Journal of Mathematical Sciences, the University of TokyoJournal of Mathematical Sciences, the University of Tokyo

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