Modern Stochastics: Theory and Applications

We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.

A complex-valued linear mixture model is considered for discrete weakly stationary processes. Latent components of interest are recovered, which underwent a linear mixing. Asymptotic properties are studied of a classical unmixing estimator which is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag τ. The main contributions are asymptotic results that can be applied to a large class of processes. In related literature, the processes are typically assumed to have weak correlations. This class is extended, and the unmixing estimator is considered under stronger dependency structures. In particular, the asymptotic behavior of the unmixing estimator is estimated for both long- and short-range dependent complex-valued processes. Consequently, this theory covers unmixing estimators that converge slower than the usual $\sqrt{T}$ and unmixing estimators that produce non-Gaussian asymptotic distributions. The presented methodology is a powerful preprocessing tool and highly applicable in several fields of statistics.

Probabilistic properties of vantage point trees are studied. A vp-tree built from a sequence of independent identically distributed points in ${[-1,\hspace{0.1667em}1]^{d}}$ with the ${\ell _{\infty }}$-distance function is considered. The length of the leftmost path in the tree, as well as partitions over the space it produces are analyzed. The results include several convergence theorems regarding these characteristics, as the number of nodes in the tree tends to infinity.

The chaos expansion of a random variable with uniform distribution is given. This decomposition is applied to analyze the behavior of each chaos component of the random variable $\log \zeta $ on the so-called critical line, where ζ is the Riemann zeta function. This analysis gives a better understanding of a famous theorem by Selberg.

A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

Moment inequalities for a class of functionals of i.i.d. random fields are proved. Then rates are derived in the central limit theorem for weighted sums of such randoms fields via an approximation by m-dependent random fields.

It is shown that the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where n is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009–2011, an upper bound is obtained for the absolute constant in the Berry–Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre–Laplace theorem which has a simple and explicit form.

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477–4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360–383] under the assumption that the kernel function potentially exhibits a singular behaviour at 0. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653–677] that studied the same mathematical question for the class of Brownian semi-stationary models.

The effect that weighted summands have on each other in approximations of $S={w_{1}}{S_{1}}+{w_{2}}{S_{2}}+\cdots +{w_{N}}{S_{N}}$ is investigated. Here, ${S_{i}}$’s are sums of integer-valued random variables, and ${w_{i}}$ denote weights, $i=1,\dots ,N$. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the ${S_{i}}$ has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.