Material for : Twisted Particle Filters

Proof of Proposition 1. Under (H2) we have g := infω,xG(x) > 0. Now consider the sequence of random variables {κn ;n ≥ 1} defined by κn := νQ ω n−1(1)g −. From (2.8) it is straightforward to establish by induction the following semigroup property: for any ω ∈ Ω, and p, n ≥ 0, Qp+n = Q ω pQ θpω n . Combined with (H2), this gives κp+n = νQ ω pQ θpω n−1(1)g − = νQp−1Q θωQθ pω n−1(1)g − ≥ κpκ pω n , and so − log κp+n ≤ − log κp − log κ pω n . Furthermore, under (H2) supω,xG(x) <∞, so there exists a finite constant c such that ˆ