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Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the “prefactor” in front of the exponentially decaying term. For fixed p ∈ (1, ∞), consider independent sequences (X(n,p))_{n∈N} and (Θ_n)_{n∈N} of random vectors with Θn distributed according to the normalized cone measure on the unit l^n_2 sphere, and X(n,p) distributed according to the normalized cone measure on the unit lnp sphere. For almost every realization (θn)_{n∈N} of (Θ_n)_{n∈N}, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of X(n,p) onto θ_n, that are asymptotically exact (as the dimension n tends to infinity). Furthermore, the case when (X(n,p))_{n∈N} is replaced with (X(n,p))_{n∈N}, where X(n,p) is distributed according to the uniform (or normalized volume) measure on the unit lnp ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions (θ_n)_{n∈N} that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of n. The results on the one hand provide more accurate quantitative estimates of tail probabilities of random projections of \ell^n_p spheres than logarithmic asymptotics, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting. The proofs combine Fourier analytic and probabilistic techniques. Along the way, several results of independent interest are obtained including a simpler representation for the quenched large deviation rate function that shows that it is strictly convex, a central limit theorem for random projections under a certain family of tilted measures, and multidimensional generalized Laplace asymptotics.
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Large deviations of bivariate Gaussian extrema
Abstract We establish sharp tail asymptotics for componentwise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.
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- Award ID(s):
- 1636069
- PAR ID:
- 10308781
- Date Published:
- Journal Name:
- Queueing Systems
- Volume:
- 93
- Issue:
- 3-4
- ISSN:
- 0257-0130
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11134-019-09632-z
