Limit Theory for Online Expectile Estimation under Short and Long Range Dependence

Expectiles are tail-sensitive distributional summaries with important applications in financial risk measurement and forecasting, yet statistical theory for their online estimation from dependent data remains limited. We study a one-pass, stochastic-gradient algorithm with constant learning rate γ for estimating the marginal expectile of a stationary causal process. The recursion admits a unique stationary causal solution and geometrically forgets its initialization. Under short-range dependence, its last iterate is asymptotically Gaussian at the √γ scale, with variance governed by the population curvature and the long-run variance of the expectile score. For long-memory linear processes the scale changes to γβ-1/2ℓ(1/γ), with a Gaussian limit arising from a fractional Brownian motion integral. We further derive the joint long-memory limit for multiple expectile levels and characterize its limiting covariance structure. Various numerical experiments corroborate the predicted Gaussian approximations and regime-specific fluctuation scales.