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

325 Pages·2003·2.163 MB·English

by  Barbe P.

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Philippe
Barbe

Philippe Barbe
APPROXIMATION
of
INTEGRALS
over
ASYMPTOTIC SETS
with applications to
Statistics and Probability

Philippe Barbe
CNRS
90 Rue de Vaugirard
75006 PARIS
FRANCE
c Ph. Barbe
(cid:13)

Contents
1. Introduction 1
Notes 16
2. The logarithmic estimate 19
Notes 23
3. The basic bounds 25
1. The normal flow and the normal foliation 25
2. Base manifolds and their orthogonal leaves 36
Notes 40
4. Analyzing the leading term for some smooth sets 43
1. Quadratic approximation of τ near a dominating manifold 44
A
2. Approximation for det (t,v) 46
A
3. What should the result be? 50
5. The asymptotic formula 53
Notes 65
6. Asymptotic for sets translated towards infinity 69
Notes 77
7. Homothetic sets, homogeneous I and Laplace’s method 79
Notes 88
8. Quadratic forms of random vectors 89
1. An example with light tail distribution 89
2. An example with heavy tail distribution 93
3. Heavy tail and degeneracy 142
Notes 162
9. Random linear forms 165
1. Some results on convex sets 166
2. Example with light tails 177
3. Example with heavy tails 185
Notes 193
10. Random matrices 197
1. Random determinants, light tails 200
2. Random determinants, heavy tails 210
3. Geometry of the unit ball of M(n,R) 218
4. Norms of random matrices 229
Notes 236
11. Finite sample results for autoregressive processes 239
1. Background on autoregressive processes 239
2. Autoregressive process of order 1 243
3. Autoregressive process of arbitrary order 255
Notes 271
12. Suprema of some stochastic processes 273
1. Maxima of processes and maxima of their variances 273
2. Asymptotic expansions for the tail of the supremum of
Gaussian processes can be arbitrary bad 276
3. Maximum of nonindependent Gaussian random variables 280
4. The truncated Brownian bridge 282
5. Polar processes on boundary of convex sets 293
Notes 296
Appendix 1. Gaussian and Student tails 297
Appendix 2. Exponential map 303
References 305
Notation 311
Postface 315