The Large Deviations of Estimating Rate Functions

Abstract

Given a sequence of bounded random variables that satisfies a well known mixing condition, it is shown that empirical estimates of the rate-function for the partial sums process satisfies the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queuelength distribution at a single server queue with infinite waiting space is proved.

Direct evaluation of dynamical large-deviation rate functions using a variational ansatz

“…In this paper we present an alternative way to calculate the correction term (18), which builds upon methods designed to compute rate functions (or their SCGF Legendre duals) empirically [61,73]. This process is more involved than the computations required to evaluate (21), but has the advantage of providing a set of clear convergence criteria and statistical error bars.…”

“…To calculate these quantities we have to compute the value of the two-dimensional SCGF (22) at various points (k δq , k a ). We can do this using a simple extension of existing techniques developed to sample points on 1D SCGFs [61,73]. Following Ref.…”

Convergence of large-deviation estimators

“…As statistical estimators,λ M (k) andÎ M (a) converge pointwise to λ(k) and I(a), respectively, in the limit of infinite sample size M → ∞. Their speed of convergence was studied in [40], following previous results on overflow probabilities and bandwidth estimates of data networks [37][38][39][40]. These studies, however, consider only bounded random variables for whichλ M (k) andÎ M (a) are known to converge quickly and uniformly.…”

“…In many cases of interest, these observables involve weakly interacting components (in space or time) which can be grouped into independent or asymptotically-independent blocks. This is the basis of the block averaging method, proposed independently in the context of free energy calculations [34] and large deviation theory [37][38][39][40].…”

“…For mixing Markov chains having a finite correlation length, it can be shown that the blocks Y i become independent in the limit where n → ∞ and b → ∞ but with b growing slower than n so that K → ∞ [40]. Moreover, if the chain is ergodic, then the Y i 's become identically distributed for i large enough, so that 1 n ln e nkAn ≈ K n ln e kYi = 1 b ln e kYi .…”