Explore the vast range of titles available.

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46-57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron-Martin paths and complementary Young regularity (CYR) of the Cameron-Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itô-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hörmander theory.

The random trigonometric series ∑ n = 1 ∞ ρ n cos ( n t + ω n ) on the circle T are studied under the conditions ∑∣ ρ n ∣ 2 = ∞ and ρ n → 0, where { ω n } are independent and uniformly distributed random variables on T . They are almost surely not Fourier-Stieltjes series but determine pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed. The behavior of the partial sums of the above series is proved to be multifractal. Our theory holds on the torus T d of dimension d ⩾ 1.

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums $\\text{Kl}_{p}(a)$ , as $a$ varies over $\\mathbf{F}_{p}^{\\times }$ and as $p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R > 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically.

We study random exponential sums of the form$\\sum_{k=1}^{n}X_{k}\\,{\\rm exp}\\{i(\\lambda _{k}^{(1)}t_{1}+\\cdots +\\lambda _{k}^{(s)}t_{s})\\}$, where$\\{X_{n}\\}$is a sequence of random variables and$\\{\\lambda _{n}^{(i)}\\colon 1\\leq i\\leq s\\}$are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered$\\{X_{n}\\}$or bounded$\\{X_{n}\\}$satisfying some mixing conditions. These results generalize recent results of Weber [Math. Inequal. Appl. 3 (2000) 443-457] and Fan and Schneider [Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 193-216] in several directions. As applications we derive conditions for uniform convergence of these sums on compact sets. We also obtain random ergodic theorems for finitely many commuting measure-preserving point transformations of a probability space. Finally, we show how some of our results allow to derive the Wiener-Wintner property (introduced by Assani [Ergodic Theory Dynam. Systems 23 (2003) 1637-1654]) for certain functions on certain dynamical systems.

We propose a method allowing us to build, for various typical means generated by the action of any given irrational rotation of the circle, examples of$L^2$functions satisfying the central limit theorem (CLT). We consider for instance nonlinear means, and means along the sequence of squares. In the latter case, the circle method of Hardy-Littlewood is used. We also give an example of continuous Gaussian random Fourier series with sample paths satisfying both CLT and almost sure CLT.

We propose a method allowing us to build, for various typical means generated by the action of any given irrational rotation of the circle, examples of$L^2$functions satisfying the central limit theorem (CLT). We consider, for instance, nonlinear means, and means along the sequence of squares. In the latter case, the circle method of Hardy and Littlewood is used. We also give an example of continuous Gaussian random Fourier series with sample paths satisfying both the CLT and the almost sure CLT.

The aim of this paper is to prove that for every $p<\\frac{4}{3}$, every p-Rider set is a q-Sidon set for all $q>\\frac{p}{2-p}$. This gives some positive answers for the union problem of p-Sidon sets. We also obtain some results on the behavior of the Fourier coefficient of a measure with spectrum in a p-Rider set.

Let X be an exponentially killed Levy process on Tn, the n-dimensional torus, that satisfies a sector condition. (This includes symmetric Levy processes.) Let Fedenote the extended Dirichlet space of X. Let h ∈ Feand {hy, y ∈ Tn} denote the set of translates of h. That is, hy(·) = h(· - y). We consider the family of zero-energy continuous additive functions {N[ hy] t, (y, t) ∈ Tn× R+} as defined by Fukushima. For a very large class of random functions h we show that$J_\\rho(T^n) = \\int(\\log N_\\rho(T^n, \\varepsilon))^{1/2} d\\varepsilon < \\infty$is a necessary and sufficient condition for the family {N[ hy] t, (y, t) ∈ Tn× R+} to have a continuous version almost surely. Here Nρ(Tn, ε) is the minimum number of balls of radius ε in the metric ρ that covers Tn, where the metric ρ is the energy metric. We argue that this condition is the natural extension of the necessary and sufficient condition for continuity of local times of Levy processes of Barlow and Hawkes. Results on the bounded variation and p-variation (in t) of N[ hy] t, for y fixed, are also obtained for a large class of random functions h.