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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.

In this note, we obtain a Tauberian theorem for a class of regular lower triangular matrices operating on cosine series with coefficients tending to zero. As corollaries we obtain Tauberian theorems for weighted mean, Nörlund, and Hausdorff matrices.

Let Λ ⊂ ℝ n be a uniformly discrete set and let C Λ be the vector space consisting of all mean periodic functions whose spectrum is simple and contained in Λ. If Λ is a gentle set then for every f ∈ C Λ we have f ( x ) = O ( ω Λ ( x )) as | x | → ∞ and ω Λ ( x ) can be estimated (Theorem 4.1). This line of research was proposed by Jean-Pierre Kahane in 1957.

This note presents two versions of an interesting theorem due to Le and Zhou [4], utilizing a new condition of bounded variation type for a trigonometric series to have an asymptotic sum.

We generalize a theorem of Leindler while the series is under the condition of MVBVS∗, and investigate the relation between the best approximation of functions and its Fourier coefficients under the given \\(L_2^p\\) norm. In the second theorem, we add a new condition to get the same result as in Theorem B.

A number of new terminating series involving and are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are generalized.

Martin and Walker ((1997) J. Appl. Prob. 34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

Recent results concerning the convergence almost everywhere or divergence everywhere of Dirichlet series Σ a^sub n^n^sup it^ appeared in the literature, revealing significant differences with the case of trigonometric series Σ^a^sub n^e^sup int^. In this work, we prove in several cases the optimality of these results. We also discuss the statistical effect of a change of signs, by considering Σ±a^sub n^n^sup it^. According to the way (probabilistic or topological) this change of signs is made, the properties of the resulting series are quite different, and can also be applied to the theory of power series. [PUBLICATION ABSTRACT]