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

arXiv: https://arxiv.org/abs/2610.16030 · HTML: https://mlsanity.com/paper/2610.16030/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 9 of 9)

- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes](https://mlsanity.com/paper/2210.00953/) (2022)
- [Online Statistical Inference for Nonlinear Stochastic Approximation with Markovian Data](https://mlsanity.com/paper/2302.07690/) (2023)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [Statistical inference for Linear Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2505.19102/) (2025)
- [Convergence and concentration properties of constant step-size SGD through Markov chains](https://mlsanity.com/paper/2306.11497/) (2023)
- [Statistical Inference with Stochastic Gradient Methods under φ-mixing Data](https://mlsanity.com/paper/2302.12717/) (2023)
- [SGD with Dependent Data: Optimal Estimation, Regret, and Inference](https://mlsanity.com/paper/2601.01371/) (2026)
- [Expectile Asymptotics](https://mlsanity.com/paper/1509.06866/) (2015)

## Abstract (arXiv)

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.