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Piecewise-deterministic Markov processes

Jolanta Kazak (2013)

Annales Polonici Mathematici

Poisson driven stochastic differential equations on a separable Banach space are examined. Some sufficient conditions are given for the asymptotic stability of a Markov operator P corresponding to the change of distribution from jump to jump. We also give criteria for the continuous dependence of the invariant measure for P on the intensity of the Poisson process.

Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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, the rate...

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

We answer a question of H. Furstenberg on the pointwise convergence of the averages 1 / N n = 1 N U ( f · R ( g ) ) , where U and R are positive operators. We also study the pointwise convergence of the averages 1 / N n = 1 N f ( S x ) g ( R x ) when T and S are measure preserving transformations.

Pointwise ergodic theorems with rate and application to the CLT for Markov chains

Christophe Cuny, Michael Lin (2009)

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

Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p>1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T 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 limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe....

Properties of Wiener-Wintner dynamical systems

I. Assani, K. Nicolaou (2001)

Bulletin de la Société Mathématique de France

In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if f L p , p large enough, is a Wiener-Wintner function then, for all γ ( 1 + 1 2 p - β 2 , 1 ] , there exists a set X f of full measure for which the series n = 1 f ( T n x ) e 2 π i n ϵ n γ converges uniformly with respect to ϵ .

Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space φ has pure discrete spectrum, we describe the collection φ P of pairs of proximal tilings in φ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then φ and ψ are homeomorphic if and...

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