Download Approximation of integrals over asymptotic sets with by Barbe P. PDF

By Barbe P.

This e-book is the 1st of a bigger venture that i could attempt to entire. A moment quantity can be dedicated to the asymptotic research of multivariate integrals over small wedges and their functions. a 3rd one should still expand many of the result of the 1st volumes to the countless dimensional atmosphere, the place there are a few very likely impressive functions within the research of stochastic procedures.

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Additional resources for Approximation of integrals over asymptotic sets with applications to statistics and probability

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To fix the ideas, a typical order of magnitude of cA,M is log I(A) for the applications that we will study. Though we want to be able to localize the study of the integral to points of A near DA , we still want A to have some thickness! In particular, we do not want the main contribution in the integral to come from the thinness of A — think for instance of taking A = Γc+ǫ \ Γc for ǫ = exp(−ec ) or even smaller, and looking for asymptotics as c tends to infinity. This can be ruled out by assuming that the first exit time of the normal flow after a time τA is large enough, namely, that for all positive M , lim inf χFA (p) = +∞ .

PROPOSITION. lim u0 →∞ u∈[0,u ] 0 I[0,bu] = For any a, b both positive, the following holds, e−au p∈DA e−τA (p) |DI(p)| 1 |DI(p)| GA (p)tv, tv 2 v∈ST (0,1) −1 p πA (p) td−k−2 dt dµp (v) dMDA (p) du b(d−k−1)/2 (d−k−1)/2 d − k + 1 2 Γ ωd−k−1 × 2 a(d−k+1)/2 e−τA (p) p∈DA t 0 |DI(p)|(d−k+1)/2 det GA (p) 1/2 dMDA (p) . REMARK. 1, the integral in t is in the range [ 0, ∞), while it is in the range [ 0, eA (p, v)) in the integral at the beginning of this section. But this will not make any difference ultimately.

This last expression looks quite complicated. However, we are almost done, and the intuition goes as follows. Roughly, we want to choose DA as ΛI(A) ∩ ∂A, so that I is minimal in A over DA — but we will actually need to have a little bit of freedom for some applications and make a slightly more subtle choice. Due to the term e−u , let us concentrate on the range u = O(1). The assumption |DI(p)| tends to infinity with |p| will imply that in good situations τA expp (tv) grows very fast as a function of t, since exp(tv) is transverse to DA where τA is minimal.

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