Sharp Finite-Time Iterated-Logarithm Martingale Concentration
This work provides foundational theoretical tools for probability and statistics, with potential applications in machine learning and finance, though it is incremental in extending existing inequalities.
The paper tackles the problem of deriving concentration bounds for martingales that are uniform over finite times, extending classical Hoeffding and Bernstein inequalities, and demonstrates optimality with a matching anti-concentration inequality, resulting in a finite-time version of the law of the iterated logarithm.
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time version of the law of the iterated logarithm, and shed light on the relationship between it and the central limit theorem.