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

This paper deals with moduli of continuity for paths of random processes indexed by a general metric space Θ with values in a general metric space X . Adapting the moment condition on the increments from the classical Kolmogorov–Chentsov theorem, the obtained result on the modulus of continuity allo...

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Bibliographic Details
Published in:Journal of theoretical probability Vol. 36; no. 3; pp. 1454 - 1486
Main Authors: Krätschmer, Volker, Urusov, Mikhail
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2023
Springer Nature B.V
Subjects:
Banach spaces
Euclidean geometry
Geometry
Mathematics
Mathematics and Statistics
Metric space
Probability Theory and Stochastic Processes
Random processes
Riemann manifold
Statistics
Theorems
Tightness
Covering numbers
60G60 Secondary 60B12
Uniform tightness
Primary 60G17
Banach-valued central limit theorems
Talagrand’s chaining technique
Kolmogorov–Chentsov type theorems
60F05
ISSN:0894-9840, 1572-9230
Online Access:Get full text
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Summary:This paper deals with moduli of continuity for paths of random processes indexed by a general metric space Θ with values in a general metric space 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 X is complete. This result is universal in the sense that its applicability depends only on the geometry of the space Θ . In particular, it is always applicable if 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.
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ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-022-01207-8