By Barbe P.

This booklet is the 1st of a bigger venture that i could try and entire. A moment quantity could 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 environment, the place there are a few in all probability extraordinary purposes within the research of stochastic procedures.

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Approximation for detA(t, v ) 47 It follows that the second fundamental form of ΛI(p) ⊂ Rd is the bilinear map B : x, y ∈ Tp ΛI(p) → B(x, y) = S(p)x, y N (p) ∈ Rd ⊖ Tp ΛI(p) . 2 in Chavel (1993). We now obtain an expression for the sectional and the Ricci curvature of ΛI(p) . 2. LEMMA The sectional curvature of the surface ΛI(p) at p is K : x, y ∈ Tp ΛI(p) → K(x, y) S(p)x, x S(p)y, y − S(p)x, y |x|2 |y|2 − x, y 2 = 2 ∈ R. Its Ricci curvature is Ricc : x, y ∈ Tp ΛI(A) → Ricc(x, y) = 1 j d−1 S(p)x, y S(p)ej , ej − S(p)x, ej S(p)ej , y ∈ R, where e1 , .

1. The normal flow and the normal foliation 35 To prove the lower bound, notice first that IA ψτA (p)+s (p) = 1 for all s in 0, χFA (p) . 9, −I(x) e dx −I(A) e A 0 s χF A (p) −τA (p) e det ψτTA (p)∗ ψτA (p) (p) DI ψτA (p) (p) p∈ΛI(A) exp − s − log DI ψτA (p)+s (p) × + log DI ψτA (p) (p) ds dMΛI(A) (p) . 1, we have d log DI ψ(p, s) ds D2 I · N ,N |DI|2 = ψ(p, s) . Therefore, − log DI ψτA (p)+s (p) log DI ψτA (p) (p) 0 t s − D2 I ψ (p) dt , |DI|2 τA (p)+t and this brings the lower estimate. REMARK.

2. The normal flow and the normal foliation 37 Let dist(· , ·) denote the Riemannian distance on ΛI(A) . For every u in Np DA , consider the radius of injectivity of ΛI(A) in the direction u, eA (p, u) = sup t 0 : dist expp (tu), p = t , the exponential map being that on ΛI(A) as before. Define ΩA,p = (t, u) : u ∈ Np DA , |u| = 1 , 0 t < eA (p, u) , p ∈ DA , so that p∈DA ∂ΩA,p is the set of all focal points of DA immersed in ΛI(A) . From its definition, we infer that expp (ΩA,p) coincide with the set ωA,p = { q ∈ ΛI(A) : there exists a unique minimizing geodesic through q which meets DA orthogonally at p }.

### Approximation of integrals over asymptotic sets with applications to statistics and probability by Barbe P.

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