A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes
Tóm tắt
This paper deals with moduli of continuity for paths of random processes indexed by a general metric space
$$\Theta $$
with values in a general metric space
$${{\mathcal {X}}}$$
. 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
$${{\mathcal {X}}}$$
is complete. This result is universal in the sense that its applicability depends only on the geometry of the space
$$\Theta $$
. In particular, it is always applicable if
$$\Theta $$
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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