By Daniel W. Stroock
This publication goals to bridge the space among likelihood and differential geometry. It provides buildings of Brownian movement on a Riemannian manifold: an extrinsic one the place the manifold is learned as an embedded submanifold of Euclidean area and an intrinsic one in line with the "rolling" map. it's then proven how geometric amounts (such as curvature) are mirrored through the habit of Brownian paths and the way that habit can be utilized to extract information regarding geometric amounts. Readers must have a powerful heritage in research with easy wisdom in stochastic calculus and differential geometry. Professor Stroock is a highly-respected specialist in likelihood and research. The readability and magnificence of his exposition extra improve the standard of this quantity. Readers will locate an inviting advent to the research of paths and Brownian movement on Riemannian manifolds.
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Additional info for An Introduction to the Analysis of Paths on a Riemannian Manifold (Mathematical Surveys and Monographs)
Sample text
S Fl and F2 belong to the same type if there are constants a > 0 and b such that F2 (x) = Fi(ax + b). 1. 5. The set of all nondegenerate distributions for suitably centered and normalized maxima consists only of distributions that belong to the types A(x), 4),,(x), and `yn(x), a > 0. 6. 's (%, (and, similarly,',) determine different types of distributions. For the proof of the theorem we need several lemmas. The following lemma is due to Khinchine. 7. f 's Fn converge weakly to a nondegenerate d.
M=n-k+l k=1,2,... 4) enables us to study the asymptotic behavior of order statistics Zk,n using well-known results for sums of independent random variables. , Petrov (1987)): if X1, X2, ... are independent random variables with expectations al, a2, ... , vari- ances vi, v, ... f. Before applying the Lyapunov inequality in our case, we will 8. f. 3) sup x n l {Zk,n-log\n-k, Ar(1) + ... + Ar(n) 1