Empirical concentration for chains with infinite memory

Concentration inequalities for time series usually contain unknown quantities that measure dependence, which stops them being used directly as confidence bounds or as criteria for learning algorithms. Recent work made such bounds empirical for stationary Markov chains by estimating those quantities. Alquier and Doukhan extend the approach to stationary chains with infinite memory.

They first prove a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. They then estimate the dependence coefficients from a single trajectory, by transition counts when the state space is finite and by derivatives of kernel conditional-density estimates when it is continuous, with error bounds that depend on the tail of the memory. A fixed multiplicative safety factor turns the estimates into empirical concentration bounds.

The procedure adapts without knowing the constants in its regularity assumptions, and the resulting confidence bounds are asymptotically as tight as the theoretical ones.