Quenched large deviations for randomly weighted geodesic random walks

where

with Λ(t) = log E(e tX 1 ).

In [3], large deviations were studied for weighted random walks. More precisely, given unit vectors θ n ∈ R n , consider the random variables

It is then shown that for almost all sequences {θ n } n of weights (with respect to the measure σ as defined in Section 2.1.2), we have

Here,

where Ψ(t) = E(Λ(tZ)) with Z ∼ N (0, 1). In particular, the rate function I w is independent of the weights θ n . This shows that for most weights, the associated weighted random walks have the same large deviations. Moreover, we see that the rate function in Cramér’s theorem is different from this universal rate function I w . As argued in [3], this shows that Cramér’s theorem is in a sense ‘atypical’.

Our aim is to extend the result from [3] to weighted geodesic random walks in Riemannian manifolds. Geodesic random walks are piecewise geodesic paths, where the directions of the geodesics are chosen at random. The weights then determine the time for which we follow each direction. We refer to Section 2.1 for a detailed description.

In [9] it was shown how Cramér’s theorem for general vector spaces can be extended to geodesic random walks. Upon analysing the proof in [3], one realizes that the result remains valid in R d , and ultimately in a general vector space V , see Theorem 2.3 for a precise statement. This opens up the possibility to apply techniques from [9] to study large deviations for weighted geodesic random walks based on the results in [3]. A key step in this methodology is the splitting of the geodesic random walks in smaller pieces. Where in [9] these pieces are identically distributed, because of the weights this is no longer the case. We overcome this by using the symmetries of the measure σ on the weights (see Section 2. 1.2) showing that each piece of the weighted geodesic random walk still follows the same large deviations, which is ultimately what we need. Furthermore, our work demonstrates that the approach in [9] is rather robust, and emphasizes the relevant properties of the stochastic processes for the methodology to work. In particular, it motivates that the methods in [9] can be extended to a general framework to study large deviations for discrete-time processes in Riemannian manifolds from their Euclidean counterparts. This can for instance be used to obtain a Riemannian analogue of the Gartner-Ellis theorem (see [2,Section 2.3]) and large deviations for Markov chains with values in Riemannian manifolds. Furthermore, it can be used to extend recent results on large deviations for random projections of l p -balls ( [4,1,8]) and projections on finite-dimensional subspaces ( [5]) to Riemannian manifolds. This will be the topic of future work.

The paper is structured as follows. In Section 2 we define weighted geodesic random walks and state our main theorem (Theorem 2.2). Furthermore, we formulate the extension of large deviation for weighted random walks as in [3] to general vector spaces. In Section 3 we prove Theorem 2.2. Following the ideas from [9], the proof is split in two parts, proving the upper bound and lower bound for the large deviation principle for the weighted geodesic random walks separately.

In a manifold, we cannot define random walks as sums of random variables. Instead, geodesic random walks are defined recursively by following pieces of geodesics (see e.g. [7,9]). We then introduce the weights as the time for which we follow each piece of geodesic. The procedure of following geodesics is encoded by the Riemannian exponential map. The map Exp :

, where γ : [0, 1] → M is the geodesic with γ(0) = x and γ(0) = v. Since we assume M is complete, Exp is defined on all of T M . With this notation at hand, we define weighted geodesic random walks.

Definition 2.1 (Weighted geodesic random walks). Fix

) is called a weighted geodesic random walk with increments {X k } 1≤k≤n and weights α n , and started at x 0 , if the following hold:

We will consider the increments {X n } n≥1 to be random variables. Note that the tangent space from which the next increment is drawn depends on the current position of the geodesic random walk. Therefore, we will put a collection {µ x } x∈M of probability measures on the tangent bundle, where µ x ∈ P(T x M ) is a probability measure on T x M for every x ∈ M .

To compare probability distributions on different tangent spaces, we identify tangent spaces at different points using parallel transport. For x, y ∈ M and a curve γ connecting x and y, we denote by τ γ;xy : T x M → T y M parallel transport along γ. If γ is a shortest geodesic between x and y, we simply write τ xy , omitting the reference to γ. Generally, we only use this notation when x and y are sufficie