Consider the standard, one dimensional, nonlinear filtering problem for diffusion processes observed in small additive white noise: dXt = b(Xt)dt + dBt , dY ε t = γ(Xt)dt + εdVt , where B·, V· are standard independent Brownian motions. Denote by qε 1(·) the density of the law of Ξ1 conditioned on σ(Y ε t : 0 ≤ t ≤ 1). We provide “quenched” large deviation estimates for the random family of meas...

The paper presents a concept of a coloring — an extension of deterministic parity automata. A coloring K is a function A∗ → N satisfying ∀α∈Aω lim inf n→∞ K(α n) <∞. Every coloring defines a subset of A by the standard parity condition [K] = { α ∈ A : lim inf n→∞ K(α n) ≡ 0 mod 2 } . We show that sets defined by colorings are exactly all ∆3 sets in the standard product topology on A. Furthermor...

• Let Ak denote the set of methods that observe a sequence of data pairs Y t = (F (θ, X ), F (τ , X )), 1 ≤ t ≤ k, and return an estimate θ̂(k) ∈ Θ. • Let FG denote the class of functions we want to optimize, where for each (F, P ) ∈ FG the subgradient g(θ;X) satisfies EP [‖g(θ;X)‖2∗] ≤ G. • For each A ∈ Ak and (F, P ) ∈ FG, consider the optimization gap: k(A, F, P,Θ) := f (θ̂(k))− inf θ∈Θ f (θ) ...

Here x0 ≥ 0 is the initial capital, c > 0 is the premium income per unit of time, N = {Nt, t ≥ 0} is an homogeneous Poisson process with rate λ and {Rk, k = 1, 2, . . .} is a sequence of i.i.d. random variables independent ofN . Henceforth we suppose that R1 has exponential distribution with mean 1/r. Our goal is to calculate explicit expressions for the distributions of the random variables su...

In this paper we are concerned with the moderate deviation principle for the normalized partial sums process Wn, considered as an element of D([0, 1]) (functions on [0, 1] with left-hand limits and continuous from the right), equipped with the Skorohod topology (see Section 14 in Billingsley (1968) for the description of the topology on D([0, 1])). More exactly, we say that the family of random...