A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes

Volker Krätschmer () and Mikhail Urusov ()
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Volker Krätschmer: University Duisburg-Essen
Mikhail Urusov: University Duisburg-Essen

Journal of Theoretical Probability, 2023, vol. 36, issue 3, 1454-1486

Abstract: Abstract This paper deals with moduli of continuity for paths of random processes indexed by a general metric space $$\Theta $$ Θ with values in a general metric space $${{\mathcal {X}}}$$ X . Adapting the moment condition on the increments from the classical Kolmogorov–Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric space $${{\mathcal {X}}}$$ X is complete. This result is universal in the sense that its applicability depends only on the geometry of the space $$\Theta $$ Θ . In particular, it is always applicable if $$\Theta $$ Θ is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result, a criterion is presented to guarantee uniform tightness of random processes with continuous paths. This is applied to find central limit theorems for Banach-valued random processes.

Keywords: Kolmogorov–Chentsov type theorems; Covering numbers; Talagrand’s chaining technique; Uniform tightness; Banach-valued central limit theorems; Primary 60G17; 60G60 Secondary 60B12; 60F05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01207-8

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