Modern Stochastics: Theory and Applications

We study convexity properties of the Rényi entropy as function of $\alpha >0$ on finite alphabets. We also describe robustness of the Rényi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but the number of events increases to ∞ and prove that the limit of Rényi entropy of the binomial distribution is equal to Rényi entropy of the Poisson distribution.

A linear structural regression model is studied, where the covariate is observed with a mixture of the classical and Berkson measurement errors. Both variances of the classical and Berkson errors are assumed known. Without normality assumptions, consistent estimators of model parameters are constructed and conditions for their asymptotic normality are given. The estimators are divided into two asymptotically independent groups.

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

In this paper, we deal with an Ornstein–Uhlenbeck process driven by sub-fractional Brownian motion of the second kind with Hurst index $H\in (\frac{1}{2},1)$. We provide a least squares estimator (LSE) of the drift parameter based on continuous-time observations. The strong consistency and the upper bound $O(1/\sqrt{n})$ in Kolmogorov distance for central limit theorem of the LSE are obtained. We use a Malliavin–Stein approach for normal approximations.

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as $d{X_{t}}=\theta (\mu +{X_{t}})dt+d{B_{t}^{a,b}}$, $t\ge 0$, with unknown parameters $\theta >0$, $\mu \in \mathbb{R}$ and $\alpha :=\theta \mu $, whereas ${B^{a,b}}:=\{{B_{t}^{a,b}},t\ge 0\}$ is a weighted fractional Brownian motion with parameters $a>-1$, $|b|<1$, $|b|<a+1$. Least square-type estimators $({\widetilde{\theta }_{T}},{\widetilde{\mu }_{T}})$ and $({\widetilde{\theta }_{T}},{\widetilde{\alpha }_{T}})$ are provided, respectively, for $(\theta ,\mu )$ and $(\theta ,\alpha )$ based on a continuous-time observation of $\{{X_{t}},\hspace{2.5pt}t\in [0,T]\}$ as $T\to \infty $. The strong consistency and the joint asymptotic distribution of $({\widetilde{\theta }_{T}},{\widetilde{\mu }_{T}})$ and $({\widetilde{\theta }_{T}},{\widetilde{\alpha }_{T}})$ are studied. Moreover, it is obtained that the limit distribution of ${\widetilde{\theta }_{T}}$ is a Cauchy-type distribution, and ${\widetilde{\mu }_{T}}$ and ${\widetilde{\alpha }_{T}}$ are asymptotically normal.