Finite-Sample Bounds for Expected Signature Estimation under Weak Dependence

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.