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Images of Gaussian random fields: Salem sets and interior points

Narn-Rueih Shieh, Yimin Xiao (2006)

Studia Mathematica

Let X = X ( t ) , t ∈ ℝ N be a Gaussian random field in ℝ d with stationary increments. For any Borel set E ⊂ ℝ N , we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Implicit Markov kernels in probability theory

Daniel Hlubinka (2002)

Commentationes Mathematicae Universitatis Carolinae

Having Polish spaces 𝕏 , 𝕐 and ℤ we shall discuss the existence of an 𝕏 × 𝕐 -valued random vector ( ξ , η ) such that its conditional distributions K x = ℒ ( η ∣ ξ = x ) satisfy e ( x , K x ) = c ( x ) or e ( x , K x ) ∈ C ( x ) for some maps e : 𝕏 × ℳ 1 ( 𝕐 ) → ℤ , c : 𝕏 → ℤ or multifunction C : 𝕏 → 2 ℤ respectively. The problem is equivalent to the existence of universally measurable Markov kernel K : 𝕏 → ℳ 1 ( 𝕐 ) defined implicitly by e ( x , K x ) = c ( x ) or e ( x , K x ) ∈ C ( x ) respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the ( e , c ) - or ( e , C ) -problem and illustrate...

Infinite ergodic index ℤ d -actions in infinite measure

E. Muehlegger, A. Raich, C. Silva, M. Touloumtzis, B. Narasimhan, W. Zhao (1999)

Colloquium Mathematicae

We construct infinite measure preserving and nonsingular rank one ℤ d -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving ℤ d -actions; for these we show that the individual basis transformations have conservative ergodic Cartesian...

Infinite Iterated Function Systems: A Multivalued Approach

K. Leśniak (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that a compact family of bounded condensing multifunctions has bounded condensing set-theoretic union. Compactness is understood in the sense of the Chebyshev uniform semimetric induced by the Hausdorff distance and condensity is taken w.r.t. the Hausdorff measure of noncompactness. As a tool, we present an estimate for the measure of an infinite union. Then we apply our result to infinite iterated function systems.

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