Sieving random iterative function systems

It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it is possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is c\`adl\`ag and has finite total variation. We also provide examples and analyse various properties of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and random continued fractions.Comment: 36 pages, 2 figures; accepted for publication in Bernoull

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