Stein's method is used to prove approximations in total variation to the distributions of integer valued random variables by (possibly signed) compound Poisson measures. For sums of independent random variables, the results obtained are very explicit, and improve upon earlier work of Kruopis (1983) and Čekanavičius (1997); coupling methods are used to derive concrete expressions for the error bounds. An example is given to illustrate the potential for application to sums of dependent random variables. ...

Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p>1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds−μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under...

Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897–923] introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the regeneration method of Nummelin, extended to the time-continuous...

Take a centered random walk $S_n$ and consider the sequence of its partial sums $A_n:={\sum }_{i=1}^n{S}_i$. Suppose $S_1$ is in the domain of normal attraction of an $\alpha$-stable law with $1lt;\alpha \le 2$. Assuming that $S_1$ is either right-exponential (i.e. $\mathbb{P}(S_{1}gt;x|S_{1}gt;0)={\mathrm{e}}^{-ax}$ for some $agt;0$ and all $xgt;0$) or right-continuous (skip free), we prove that $$\mathbb{P}\{A_{1}gt;0,\cdots ,A_{N}gt;0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$ as $N\to \infty$, where $C_{\alpha }gt;0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially...

Let $f\in \mathbb{Z}\left[x\right]$ be a polynomial of degree $d\ge 3$ without roots of multiplicity $d$ or $(d-1)$. Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f\left(p\right)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy.The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations.

We find precise small deviation asymptotics with respect to the Hilbert norm for some special Gaussian processes connected to two regression schemes studied by MacNeill and his coauthors. In addition, we also obtain precise small deviation asymptotics for the detrended Brownian motion and detrended Slepian process.

We show that a certain type of quasifinite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasifinite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with normalization ∝ √n. Lastly, we show that certain ergodic, probability preserving...