Finite-Sample Bounds for Expected Signature Estimation under Weak Dependence

TL;DR

This paper establishes finite-sample mean-squared error bounds for estimating the expected signature of a stochastic process from dependent data, applicable to rough paths with various dependence structures.

Contribution

It provides the first non-asymptotic bounds for expected signature estimation under weak dependence, including long-range dependence, using block-averaging and rough-path theory.

Findings

Error bounds separate discretization and fluctuation terms.

Rates depend on path regularity and dependence decay.

Empirical results show faster convergence than theoretical bounds.

Abstract

The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating it from a single long dependent trajectory have been lacking. We study a stationary stochastic process whose sample paths can be interpreted as geometric rough paths, partitioned into blocks of equally-spaced observations, and prove a non-asymptotic mean-squared error bound for the block-averaging estimator. Rough-path theory is required for the estimand to be well-defined when paths have H\"older regularity at most $1/2$, because Young and Riemann--Stieltjes integration cannot define the signature's iterated integrals. Under moment and stationarity assumptions together with a covariance-decay condition on block signatures -- strictly weaker than $\alpha$-mixing and applicable to long-range-dependent drivers -- the error separates into a discretization term and a fluctuation term, with rates determined respectively by path regularity and dependence strength. A level-wise rough-factorial variance analysis keeps finite-truncation constants explicit and yields an optimal allocation rule under a fixed observation budget. We verify the assumptions for fractional Ornstein--Uhlenbeck processes in three regimes, namely rough (Hurst $H<1/2$), semimartingale ($H=1/2$), and long-range ($H>1/2$). Monte Carlo experiments show empirical convergence rates faster than the theoretical upper bounds.