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On Talagrand's Admissible Net Approach to Majorizing Measures and Boundedness of Stochastic Processes

100%
Bednorz W.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
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nr 1
83-91
EN
We show that the main result of [1] on sufficiency of existence of a majorizing measure for boundedness of a stochastic process can be naturally split in two theorems, each of independent interest. The first is that the existence of a majorizing measure is sufficient for the existence of a sequence of admissible nets (as recently introduced by Talagrand [5]), and the second that the existence of a sequence of admissible nets is sufficient for sample boundedness of a stochastic process with bounded increments.
2
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Majorizing Measures and Ultrametric Spaces

100%
Bednorz W.
Bulletin of the Polish Academy of Sciences. Mathematics
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2012
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tom 60
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nr 1
91-100
EN
Talagrand's proof of the sufficiency of existence of a majorizing measure for the sample boundedness of processes with bounded increments used a contraction from a certain ultrametric space. We give a short proof of existence of such an ultrametric using admissible sequences of nets.
3
Content available remote

The Kendall theorem and its application to the geometric ergodicity of Markov chains

100%
Bednorz W.
Applicationes Mathematicae
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2013
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tom 40
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nr 2
129-165
EN
We give an improved quantitative version of the Kendall theorem. The Kendall theorem states that under mild conditions imposed on a probability distribution on the positive integers (i.e. a probability sequence) one can prove convergence of its renewal sequence. Due to the well-known property (the first entrance last exit decomposition) such results are of interest in the stability theory of time-homogeneous Markov chains. In particular this approach may be used to measure rates of convergence of geometrically ergodic Markov chains and consequently implies estimates on convergence of MCMC estimators.
4
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A Note on the Men'shov-Rademacher Inequality

100%
Bednorz W.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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tom 54
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nr 1
89-93
EN
We improve the constants in the Men'shov-Rademacher inequality by showing that for n ≥ 64, $E(sup_{1≤k≤n} |∑^k_{i=1} X_i|² ≤ 0.11(6.20 + log₂n)²$ for all orthogonal random variables X₁,..., Xₙ such that $∑^n_{k=1} E|X_k|² = 1$.
5
Content available remote

The majorizing measure approach to sample boundedness

100%
Bednorz W.
Colloquium Mathematicum
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2015
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tom 139
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nr 2
205-227
EN
We describe an alternative approach to sample boundedness and continuity of stochastic processes. We show that the regularity of paths can be understood in terms of the distribution of the argument maximum. For a centered Gaussian process X(t), t ∈ T, we obtain a short proof of the exact lower bound on $𝔼 sup_{t∈ T}X(t)$. Finally we prove the equivalence of the usual majorizing measure functional to its conjugate version.
6
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On a Sobolev type inequality and its applications

100%
Bednorz W.
Studia Mathematica
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2006
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tom 176
|
nr 2
113-137
EN
Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball $T := B_{||·||}(0,r)$, r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, $sup_{s,t∈ T} |f(s)-f(t)| ≤ 6AB(∫_{0}^{r} ψ(1/Aε^{n-1})ε^{n-1} dε + 1/(n|B_{||·||}(0,1)|) ∫_{T} φ(1/B ||∇f(u)||⁎)du)$, where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on φ so that each separable process X(t), t ∈ T, which satisfies $||X(s)-X(t)||_{φ} ≤ η(||s-t||)$ for s,t ∈ T is a.s. sample bounded.
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