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Quenched large deviations for randomly weighted geodesic random walks

Rik Versendaal

Abstract

We consider weighted geodesic random walks in a complete Riemannian manifold $(M,g)$. We show that for almost all sequences of weights (with respect to a suitable measure), these weighted geodesic random walks satisfy, when suitably scaled, a large deviation principle with a universal rate function. This extends the results from [3], where this was shown for the real-valued case. It turns out the argument is also valid for general vector spaces. This allows us to use the methodology of [9], in which large deviations for geodesic random walks are obtained from large deviation estimates for associated random walks in tangent spaces.

Quenched large deviations for randomly weighted geodesic random walks

Abstract

We consider weighted geodesic random walks in a complete Riemannian manifold . We show that for almost all sequences of weights (with respect to a suitable measure), these weighted geodesic random walks satisfy, when suitably scaled, a large deviation principle with a universal rate function. This extends the results from [3], where this was shown for the real-valued case. It turns out the argument is also valid for general vector spaces. This allows us to use the methodology of [9], in which large deviations for geodesic random walks are obtained from large deviation estimates for associated random walks in tangent spaces.
Paper Structure (14 sections, 11 theorems, 89 equations)

This paper contains 14 sections, 11 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Randomly weighted geodesic random walks
  3. Geodesic random walks with weighted increments
  4. Identically distributed increments and parallel transport
  5. Weight distribution
  6. Main result
  7. Large deviations for weighted random walks in vector spaces
  8. Connection to large deviations for $k$-dimensional projections
  9. Proof of Theorem \ref{['thm:weighted_GRW_LDP']}
  10. Notation and preliminary results
  11. The upper bound of the large deviation principle
  12. Lower bound of the large deviation principle
  13. From $T_{x_0}M$ to $M$
  14. Proof of the large deviations lower bound

Key Result

Theorem 2.2

Let $\theta \in \mathbb{S}$ and let $\{\mathcal{S}_n^{\theta^n}\}_{n \in \mathbb{N}}$ be the weighted geodesic random walk as defined above. Assume the increments of the geodesic random walks are bounded, independent and identically distributed. Let $\sigma$ be a measure on $\mathbb{S}$ as in Sectio where $\Psi(\lambda) = \mathbb{E}(\Lambda_{x_0}(Z\lambda))$ with $Z \sim N(0,1)$ and $\Lambda_{x_0}

Theorems & Definitions (22)

  • Definition 2.1: Weighted geodesic random walks
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Proposition 2.7
  • proof
  • Lemma 3.1
  • ...and 12 more