Thursday, January 23, 2025 - 16:00 in V3-201
Cramér-type moderate deviation principles, optimal transport and weak dependence
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Moritz Jirak from Universität Wien
| Abstract: |
Consider a stationary, weakly dependent sequence of random variables.
Subject to mild conditions, allowing for polynomial decay in
correlation, we show Cramér-type moderate deviation bounds with
optimal rate $n^{-1/2}$. As an application, we obtain non-uniform
Berry-Esseen bounds with an optimal moment to weight-function relation
$p \mapsto (1 + |x|)^{p}$ and rate $n^{-1/2}$. In addition, we also
derive rate-optimal bounds for the central limit theorem with respect to
optimal transport distances $\mathrm{W}_q$, $q \geq 1$, where we bypass
arguments based on higher order moment-matching like Edgeworth
expansions or Stein's method in connection with Rio's inequality. The
setup is quite general, and contains many prominent dynamical systems
and time series models, including random walks on the general linear
group and other slowly or quickly mixing dynamical systems arising from
intermittent, logistic or nonuniformly hyperbolic maps, functionals of
(augmented) Garch models of any order, functionals of dynamical systems
arising from SDEs like the Langevin diffusion, iterated random functions
and many more.
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