Functional large deviations for Cox processes and $\mathit{Cox}/G/\infty$ queues, with a biological application

Upper large deviations for power-weighted edge lengths in spatial random networks

“…which in particular implies that sup n∈N E[R(π n )] < ∞ in accordance with (12). In Lemma 6 and Lemma 7 we showed tightness of (π n , ηn ) n .…”

“…Essentially, an annealed LDP is an LDP for a sequence of marginal distributions given by a product measure that can be derived if the corresponding sequences of the other marginal probability distribution and the conditional probability distributions satisfy LDPs. The LDP for the conditional probability distribution is called quenched LDP, see [10,12].…”

“…Next, for a bounded measurable function G : P(X ) → R and every n ∈ N, we henceforth assume that (π k,n ) k is a collection of random probability measures from the infimum of Proposition 5 such that (12) sup…”

“…Under certain conditions annealed large deviations follow from the quenched large deviations. [10, Theorem 2.3] states a set of requirements that enables this transition and which has been successfully applied in the past for instance in [12]. Those conditions adapted to our setting are the following:…”

“…However, our Skorokhod setting requires a more careful analysis, where a lot of our work involves dealing with truncations, which [7] did not have to consider. The annealed LDP is derived from the quenched LDP by using a general result that can be found in [10] and [12]. The proof that a compactly supported data distribution is a sufficient condition for the uniqueness property is based on [28], who proved a very similar result but with the mentioned difference that stems from the representation, we have to deal with more measures than only the fixed data distribution.…”