On quantitative convergence for stochastic processes: Crossings, fluctuations and martingales

TL;DR

This paper introduces a unified framework for deriving uniform bounds on the stability of stochastic processes, including martingales and their generalizations, using logical and proof-theoretic methods to extract quantitative convergence information.

Contribution

It provides a novel, abstract approach to quantify convergence in stochastic processes, extending classical results like Doob's theorem to more complex processes such as almost-supermartingales.

Findings

Quantitative bounds for $L_1$-sub- and supermartingales derived.

Framework extends to complex processes like almost-supermartingales.

Proof-theoretic methods enable extraction of low-complexity quantitative data.

Abstract

We develop a general framework for extracting highly uniform bounds on local stability for stochastic processes in terms of information on fluctuations or crossings. This includes a large class of martingales: As a corollary of our main abstract result, we obtain a quantitative version of Doob's convergence theorem for $L_1$-sub- and supermartingales, but more importantly, demonstrate that our framework readily extends to more complex stochastic processes such as almost-supermartingales, thus paving the way for future applications in stochastic optimization. Fundamental to our approach is the use of ideas from logic, particularly a careful analysis of the quantifier structure of probabilistic statements and the introduction of a number of abstract notions that represent stochastic convergence in a quantitative manner. In this sense, our work falls under the 'proof mining' program, and indeed, our quantitative results provide new examples of the phenomenon, recently made precise by the first author and Pischke, that many proofs in probability theory are proof-theoretically tame, and amenable to the extraction of quantitative data that is both of low complexity and independent of the underlying probability space.