Modern Stochastics: Theory and Applications

A novel theoretical result on estimation of the local time and the occupation time measure of an α-stable Lévy process with $\alpha \in (1,2)$ is presented. The approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an ${L^{2}}$-optimal statistic by construction. The corresponding stable central limit theorems are proved and a statistical application is discussed. In particular, this work extends the results of [20], which investigated the case of the Brownian motion.

Let $({\xi _{1}},{\eta _{1}})$, $({\xi _{2}},{\eta _{2}}),\dots $ be independent identically distributed ${\mathbb{N}^{2}}$-valued random vectors with arbitrarily dependent components. The sequence ${({\Theta _{k}})_{k\in \mathbb{N}}}$ defined by ${\Theta _{k}}={\Pi _{k-1}}\cdot {\eta _{k}}$, where ${\Pi _{0}}=1$ and ${\Pi _{k}}={\xi _{1}}\cdot \dots \cdot {\xi _{k}}$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets $\{{\Pi _{1}},{\Pi _{2}},\dots ,{\Pi _{k}}\}\subset \mathbb{N}$ and $\{{\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{k}}\}\subset \mathbb{N}$, $k\in \mathbb{N}$, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.

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. The notion of noncentral moderate deviations is used when the weak convergence is towards a non-Gaussian distribution. In this paper, noncentral moderate deviation results are presented for two fractional Skellam processes known in the literature (see [20]). It is established that, for the fractional Skellam process of type 2 (for which one can refer to the recent results for compound fractional Poisson processes in [3]), the convergences to zero are usually faster because one can prove suitable inequalities between rate functions.

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.