Modern Stochastics: Theory and Applications

We establish the rate of convergence of distributions of sums of independent identically distributed random variables to the Gaussian distribution in terms of truncated pseudomoments by implementing the idea of Yu. Studnyev for getting estimates of the rate of convergence of the order higher than ${n}^{-1/2}$.

For a (non-symmetric) strong Markov process X, consider the Feynman–Kac semigroup where A is a continuous additive functional of X associated with some signed measure. Under the assumption that X admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to ${T_{t}^{A}}$ possesses the density ${p_{t}^{A}}(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for ${p_{t}^{A}}(x,y)$. Some examples are provided.

We consider the Berkson model of logistic regression with Gaussian and homoscedastic error in regressor. The measurement error variance can be either known or unknown. We deal with both functional and structural cases. Sufficient conditions for identifiability of regression coefficients are presented.

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model where ${B}^{H_{1}}$ and ${B}^{H_{2}}$ are two independent fractional Brownian motions with Hurst indices $H_{1}$ and $H_{2}$ satisfying the condition $\frac{1}{2}\le H_{1}<H_{2}<1$. Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence–uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

In this paper, we provide strong $L_{2}$-rates of approximation of the integral-type functionals of Markov processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its transition probability density, we get the accuracy that coincides with that obtained in [3] for a one-dimensional diffusion process.

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter G-time-changed Brownian motions. In addition, we study the “sequence-independent variation” property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.