Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces

G. Krupa

Displaying similar documents to “Ergodic theorems for subadditive superstationary families of random sets with values in Banach spaces”

Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

Studia Mathematica

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Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$ -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$ -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these...

Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards

Francis Comets, Serguei Popov (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a random walk in a stationary ergodic environment in $ℤ$ , with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no “traps.” We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in $ℝ^{d}$ , $d \geq 3$ , which serves...

Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson, T. A. Gillespie (2001)

Studia Mathematica

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Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ (U)}_{n = 1}^{\infty}$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ (U) = 1 / n \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) U^{k}$ . We show that this sequence ${ₙ (U)}_{n = 1}^{\infty}$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral...

Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_{k}$ such that the weighted ergodic averages corresponding to $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...

Ergodic transforms associated to general averages

H. Aimar, A. L. Bernardis, F. J. Martín-Reyes (2010)

Studia Mathematica

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Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^{p}$ , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^{p}$ . For p = 1 we find that the...

Vector-valued ergodic theorems for multiparameter Additive processes II

Ryotaro Sato (2003)

Colloquium Mathematicae

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Previously we obtained stochastic and pointwise ergodic theorems for a continuous d-parameter additive process F in L₁((Ω,Σ,μ);X), where X is a reflexive Banach space, under the condition that F is bounded. In this paper we improve the previous results by considering the weaker condition that the function $W (\cdot) = e s s s u p {| | F (I) (\cdot) | | : I \subset [0, 1)}^{d}$ is integrable on Ω.

Random fixed points of increasing compact random maps

Ismat Beg (2001)

Archivum Mathematicum

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Let $(Ω, Σ)$ be a measurable space, $(E, P)$ be an ordered separable Banach space and let $[a, b]$ be a nonempty order interval in $E$ . It is shown that if $f : Ω \times [a, b] \to E$ is an increasing compact random map such that $a \leq f (ω, a)$ and $f (ω, b) \leq b$ for each $ω \in Ω$ then $f$ possesses a minimal random fixed point $α$ and a maximal random fixed point $β$ .

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

Studia Mathematica

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...

JOP's counting function and Jones' square function

Karin Reinhold (2006)

Studia Mathematica

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We study a class of square functions in a general framework with applications to a variety of situations: samples along subsequences, averages of $ℤ ₊^{d}$ actions and of positive L¹ contractions. We also study the relationship between a counting function first introduced by Jamison, Orey and Pruitt, in a variety of situations, and the corresponding ergodic averages. We show that the maximal counting function is not dominated by the square functions.

Ergodic and Mixing Random Walks on Locally Compact Groups.

Joseph Rosenblatt (1981)

Mathematische Annalen

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Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends, Jörg Schmeling (2001)

Czechoslovak Mathematical Journal

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We call a sequence $(T_{n})$ of measure preserving transformations strongly mixing if $P (T_{n}^{- 1} A \cap B)$ tends to $P (A) P (B)$ for arbitrary measurable $A$ , $B$ . We investigate whether one can pass to a suitable subsequence $(T_{n_{k}})$ such that $\frac{1}{K} \sum_{k = 1}^{K} f (T_{n_{k}}) ⟶ \int f d P$ almost surely for all (or “many”) integrable $f$ .

Vector-valued ergodic theorems for multiparameter additive processes

Ryotaro Sato (1999)

Colloquium Mathematicae

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Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T=T(u):u=( $u_{1}$ , ... , $u_{d})$ , $u_{i}$ ≥ 0, 1 ≤ i ≤ d be a strongly measurable d-parameter semigroup of linear contractions on $L_{1}$ ((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_{1}$ ((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_{1}$ ((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems...