We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$-measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$-measure.

In the limit theory for strictly stationary processes $f\circ T^i,i\in \mathbb{Z}$, the decomposition $f=m+g-g\circ T$ proved to be very useful; here $T$ is a bimeasurable and measure preserving transformation an $(m\circ T^i)$ is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of $(m\circ T^i)$ is fixed. The case when the filtration varies is solved in [13]. The necessary and sufficient condition of the existence of the decomposition were given in [12] (for earlier and weaker versions of the results see [7])....

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-optimal quantizers of an ${ℝ}^d$-valued random vector $X\in L^r\left(\mathbb{P}\right)$ defined in the probability space $(\Omega ,𝒜,\mathbb{P})$ with distribution ${\mathbb{P}}_X=P$. To be precise, we investigate the Ls-quantization rate of sequences ${\alpha }_n^{\theta ,\mu }=\mu +\theta ({\alpha }_n-\mu )=\{\mu +\theta (a-\mu ),a\in {\alpha }_n\}$ when $\theta \in {ℝ}_{+}^{☆},\mu \in ℝ,s\in (0,r)$ or s ∈ (r, +∞) and $X\in L^s\left(\mathbb{P}\right)$. We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls-rate-optimal. For the Gaussian and the exponential distributions we show...

A simple renewal process is a stochastic process $\left\{X_n\right\}$ taking values in $\{0,1\}$ where the lengths of the runs of $1$’s between successive zeros are independent and identically distributed. After observing $X_0,X_1,...X_n$ one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional distribution...

We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha$-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta \left(\alpha \right)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier...

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $Hgt;1/3$. We show that under some geometric conditions, in the regular case $Hgt;1/2$, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $Hgt;1/3$ and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper...