Volume 219, Nº 5 (2016)

In this paper, we continue the study of the class of hypergeometric diffusions started by the author. A broad subclass of these diffusions consists of hyperbolic Ornstein–Uhlenbeck processes. An explicit formula for the transition density of a hyperbolic Ornstein–Uhlenbeck process is derived. Bibliography: 7 titles.

In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain’s entropy criteria, and Kakutani–Rokhlin lemma, the most classical device for these questions in ergodic theory. First, we state a L¹-version of the continuity principle and give an example of its usefulness by applying it to a famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in L^p, 2 ≤ p ≤ ∞, and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on L². We further study the corresponding criterion in L^p, 1 < p < 2, using properties of pstable processes. Finally, we consider Kakutani–Rokhlin’s lemma, one of the most frequently used tools in ergodic theory, by stating and proving a criterion for a.e. divergence of weighted ergodic averages. Bibliography: 38 titles.

Consider a random polynomial G_Q(x) = ξ_Q,nxⁿ + ξ_{Q,n − 1}x^n − 1 + ⋯ + ξ_Q,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, . . .,Q}. Denote by D(G_Q) the discriminant of G_Q. We show that there exists a constant C_n depending on n only such that for all Q ≥ 2, the distribution of D(G_Q) can be approximated as follows: \( \underset{-\infty \le a\le b\le -\infty }{ \sup}\left|\mathrm{P}\left(a\frac{D\left({G}_Q\right)}{Q^{2n-2}}\le b\right)-{\displaystyle \underset{a}{\overset{b}{\int }}{\upvarphi}_n(x)dx}\right|\le \frac{C_n}{ \log Q}, \) where \( \varphi \)_n denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(G_Q) denote the minimal distance between complex roots of G_Q. As an application, we show that for any ε > 0 there exists a constant δ_n > 0 such that Δ(G_Q) is stochastically bounded from below/above for all sufficiently large Q in the following sense: \( \mathrm{P}\left({\delta}_n<\varDelta \left({G}_Q\right)<\frac{1}{\delta_n}\right)>1-\varepsilon \). Bibliography: 14 titles.

Parametrized families of distributions for the circular unitary ensemble in random matrix theory are considered; they are connected to Toeplitz determinants and have many applications in mathematics (for example, to the longest increasing subsequences of random permutations) and physics (for example, to nuclear physics and quantum gravity). We develop a theory for the unknown parameter estimated by an asymptotic maximum likelihood estimator, which, in the limit, behavesas the maximum likelihood estimator if the latter is well defined and the family is sufficiently smooth. They are asymptotically unbiased and normally distributed, where the norming constants are unconventional because of long range dependence.

In this paper, we study a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric, infinitely divisible distributions. It is shown that the values at zero of the concentration functions of weighted sums of i.i.d. random variables may be estimated by the values at zero of the concentration functions of symmetric, infinitely divisible distributions with the Lévy spectral measures which are multiples of the sum of delta-measures at ±weights involved in constructing the weighted sums.

We consider a field f \( \circ {T}_1^{i_1}\circ \dots \circ {T}_d^{i_d} \) , where T₁, . . . , T_d are completely commuting transformations in the sense of Gordin. If one of these transformations is ergodic, we give sufficient conditions in the spirit of Hannan under which the partial sum process indexed by quadrants converges in distribution to a Brownian sheet. The proof combines a martingale approximation approach with a recent CLT for martingale random fields due to Volný. We apply our results to completely commuting endomorphisms of the m-torus. In that case, the conditions can be expressed in terms of the L²-modulus of continuity of f.

In this note, we show that a stationary sequence obtained by applying a fixed deterministic function to shifts of a stationary sequence (satisfying a mild regularity condition) has a spectral density. In the multiparametric setting, we obtain a similar result for a function of a shifted i.i.d. field.

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact, separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e., energy forms on finite weighted graphs, in the sense of Mosco convergence, i.e., strong resolvent convergence.