Summary
LetX i,iɛN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and Φ a mappingB →R. 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\({{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} {Z_n }}} \right. \kern-\nulldelimiterspace} {Z_n }}\).
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Bolthausen, E. Laplace approximations for sums of independent random vectors. Probab. Th. Rel. Fields 72, 305–318 (1986). https://doi.org/10.1007/BF00699109
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DOI: https://doi.org/10.1007/BF00699109