Probability Theory and Related Fields

, Volume 72, Issue 2, pp 305–318

# Laplace approximations for sums of independent random vectors

• E. Bolthausen
Article

## Summary

LetXi,iɛN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and Φ a mappingBR. Under some conditions an asymptotic evaluation of$$Z_n = E\left( {\exp \left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} \right)$$ is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums$$\sum\limits_{i = 1}^n {X_i }$$ under the law transformed by the density exp$${{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} {Z_n }}} \right. \kern-\nulldelimiterspace} {Z_n }}$$.

## Keywords

Banach Space Stochastic Process Probability Theory Limit Theorem Random Vector
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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