Abstract: Let B be a uniformly convex Banach space, let T be a nonexpansive linear operator, and ... more Abstract: Let B be a uniformly convex Banach space, let T be a nonexpansive linear operator, and let A_n x denote the ergodic average (1/n) sum_ {i< n} T^ n x. A generalization of the mean ergodic theorem due to Garrett Birkhoff asserts that the sequence (A_n x) converges, which is equivalent to saying that for every epsilon> 0, the sequence has only finitely many fluctuations greater than epsilon.
Abstract: The dominated convergence theorem implies that if (fn) is a sequence of functions on a ... more Abstract: The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise ae, then (∫ fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (fn) and the underlying space.
Computable randomness and betting for computable probability spaces
Abstract: Unlike Martin-L\" of randomness and Schnorr randomness, computable randomness has not b... more Abstract: Unlike Martin-L\" of randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, many equivalent definitions), and further, provides a general method for abstracting" bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces.
Algorithmic randomness, reverse mathematics, and the dominated convergence theorem
We analyze the pointwise convergence of a sequence of computable elements of L1 (2ω) in terms of ... more We analyze the pointwise convergence of a sequence of computable elements of L1 (2ω) in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set.
Abstract: We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness... more Abstract: We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a" straightforward modification of the proof of van Lambalgen's Theorem." This is not so, and we point out why.
• Pathak proved the Effective LDT for Martin-Löf randoms.• Brattka, Miller, Nies characterized co... more • Pathak proved the Effective LDT for Martin-Löf randoms.• Brattka, Miller, Nies characterized computable randomness, Martin-Löf randomness, and weak-2 randomness in terms of differentiability of absolutely-continuous computable functions on [0, 1]. Martin-Löf case was based on the work of Demuth.• Pathak, Rojas, Simpson have a different proof of the Effective LDT for Schnorr randoms.
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Papers by Jason Rute