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We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^p(X,\Sigma ,\mu )$, p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

Let X be a closed subspace of $L^p\left(\mu \right)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $su{p}_{n\in \mathbb{Z}}\left|\right|Uⁿ\left|\right|<\infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like ${\sum }_{n\ge 1}\left(Uⁿf\right)/n^{1-\alpha }$, 0 ≤ α < 1, in terms of $\left|\right|f+\cdots +U^{n-1}f{\left|\right|}_p$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie....

We give a proof of convergence of moments in the Central Limit Theorem (under the Lyapunov-Lindeberg condition) for triangular arrays, yielding a new estimate of the speed of convergence expressed in terms of νth moments. We also give an application to the convergence in the mean of the pth moments of certain random trigonometric polynomials built from triangular arrays of independent random variables, thereby extending some recent work of Borwein and Lockhart.

Let be Dunford–Schwartz operator on a probability space (, ). For ∈ (), &gt;1, we obtain growth conditions on ‖∑ ‖ which imply that (1/ )∑ →0 -a.e. In the particular case that =2 and is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central...

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.

Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform $li{m}_n{\sum }_{k=1}^n\left(T^{k}x\right)/k$. We prove that weak and strong convergence are equivalent, and in a reflexive space also $su{p}_n\left|\right|{\sum }_{k=1}^n\left(T^{k}x\right)/k\left|\right|<\infty$ is equivalent to the convergence. We also show that $-{\sum }_{k=1}^{\infty }\left(T^k\right)/k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup ${(I-T)}_{|\overline{(I-T)X}}^r$.