On the number of empty boxes in the Bernoulli sieve

The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1...W_{k-1}(1-W_k)$, where (W_k)_{k\in\mn} are independent copies of a random variable $W$ taking values in $(0,1)$. Assuming that the number of balls equals $n$, let $L_n$ denote the number of empty boxes within the occupancy range. The paper proves that, under a regular variation assumption, $L_n$, properly normalized without centering, weakly converges to a functional of an inverse stable subordinator. Proofs rely upon the observation that $(\log P_k)$ is a perturbed random walk. In particular, some results for general perturbed random walks are derived. The other result of the paper states that whenever $L_n$ weakly converges (without normalization) the limiting law is mixed Poisson.Comment: Minor corrections to Proposition 5.1 were adde

Local universality for real roots of random trigonometric polynomials

Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$X_n(t) = \sum_{k=1}^n \left( \xi_k \sin (kt) + \eta_k \cos (kt)\right),$$ where $(\xi_1,\eta_1),(\xi_2,\eta_2),\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as $n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correlation function $(\sin t)/t$. We also establish similar local universality results for the centered random vectors $(\xi_k,\eta_k)$ having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional $\alpha$-stable law.Comment: 20 pages, extended version. New results (including the stable case) were adde

A functional limit theorem for general shot noise processes

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally H\"{o}lder continuous Gaussian limit process and that the response function is regularly varying at infinity we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann-Liouville process. We specialize our result for five particular counting processes. Also, we investigate H\"{o}lder continuity of the limit processes for general shot noise processes.Comment: 15 pages, submitted to a journa

Exponential moments of first passage times and related quantities for random walks

For a zero-delayed random walk on the real line, let $\tau(x)$, $N(x)$ and $\rho(x)$ denote the first passage time into the interval $(x,\infty)$, the number of visits to the interval $(-\infty,x]$ and the last exit time from $(-\infty,x]$, respectively. In the present paper, we provide ultimate criteria for the finiteness of exponential moments of these quantities. Moreover, whenever these moments are finite, we derive their asymptotic behaviour, as $x \to \infty$