How to estimate the rate function of a cumulative process

Abstract

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.

The Large Deviations of Estimating Rate Functions

“…Having established the LDP for the CGF estimates in Theorem 4.2, the rate function estimator result follows from another application of the contraction principle in conjunction with the continuity of the Legendre-Fenchel transform defined in equation (11). Considering LF : X Λ → X I , the map LF is a homeomorphism [4,5] and, indeed, this is in part what leads us to this topology in order to establish these results; it correctly captures smoothness in convex conjugation.…”

“…Therefore d ((x 3 , y 3 ), epi(f )) = lim n→∞ d ((x 3 , y 3 ), epi(f n )) = 0, and so (x 3 , y 3 ) ∈ epi(f ). [11][Proposition 3], although not immediately as the map L fails that proposition's criteria. We will bypass this difficulty by considering a sequence of restrictions of L that satisfy the propositions criteria and such that the images of elements of X M under such restrictions are equal to their image under L when intersected with an increasingly large neighbourhood of the origin.…”

“…Then epi(L(f )) = g * (epi(f )), where g * : P(R × (0, ∞)) → P(R 2 ), with P denoting the power set, and g * (D) = {(θ, ψ) : g(θ, ψ) ∈ D} is the pull back. Although g is bijective, continuous and maps bounded sets to bounded sets, its inverse fails to be uniformly continuous on bounded sets, so that the assumptions of [11][Proposition 3] do not hold. Now consider g n = g| R×[−n,∞) , the function g restricted to the set R × [−n, ∞) with codomain R × [exp(−n), ∞), and the corresponding pullback g * n : P(R × [exp(−n), ∞)) → P(R × [−n, ∞)).…”

“…Then g n is bijective, continuous and with an inverse uniformly continuous on bounded subsets. Therefore by [11][Proposition 3] g * n is continuous. Now consider any sequence of closed subsets of R × (0, ∞), {A m }, all containing the point (0, 1) and converging in (P(R × (0, ∞)), τ AW ) to A, which, by properties of τ AW convergence, must also contain the point (0, 1).…”

Large deviations for the empirical measure of heavy-tailed Markov renewal processes

“…[9,10,11,12,18]. Typical techniques include sharp renewal estimates [1, Chapter XIII] and so-called contraction principles via inversion maps [18,10]. In particular one can obtain large deviations principles for the renewal version of processes whose detailed asymptotic properties are known, using inversion maps [18,10].…”