Uniform bounds under increment conditions

Abstract

Abstract. We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.

Norm convergence of some power series of operators in Lpwith applications in ergodic theory

“…We give a theorem which is a combination of results of Cohen and Lin [3] and of Weber [18] obtained for general (not necessarily invertible) power bounded operators on some fixed L p .…”

“…1. (i) and (ii) of Theorem 6.3 follow from Theorem 1.3 of [18] applied with M n = n, L(x) = x(log x) 1+ε , Ψ (x) = x p(1−α) /(log x) 1+η and ϕ(n) given by the corresponding denominator in (i) or (ii).…”

Convergence of weighted ergodic averages

“…In the last section, we provide several examples illustrating our main results. The type of questions that we deal with has already been investigated and many methods have been provided in this way (see [17] for more details). However, our methods are simpler and provide new examples.…”

“…Example 4. Here we deal with an example which was considered in [5,17]. Let (X k ) be a sequence of i.i.d random variables defined on some probability space (Ω, B, P), and let W k (ω) = e 2iπX k (ω) , ω ∈ Ω.…”

Sampling the Lindelöf hypothesis for Dirichlet $$L$$ L -functions by the Cauchy random walk

“…Hence, Lifshits and Weber showed that the average value of ζ(s) on the Cauchy random walk s = 1/2+i S n equals one, which implies that the values of the zeta-function are small on average. Their proof is based on the following proposition of Weber [7]. Proposition 1.3 Let {m l : l ≥ 1} be a sequence of positive reals with partial sums M n = n l=1 m l tending to infinity with n. Assume that…”