Open Access
2021 Stochastic approximation algorithms for superquantiles estimation
Bernard Bercu, Manon Costa, Sébastien Gadat
Author Affiliations +
Electron. J. Probab. 26: 1-29 (2021). DOI: 10.1214/21-EJP648


This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile, also known as conditional value-at-risk, estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.

Funding Statement

The authors gratefully acknowledge financial support from the Agence Nationale de la Recherche (MaSDOL grant ANR-19-CE23-0017).


The authors warmly thank the two anonymous referees for their insightful comments and constructive suggestions which helped to improve the paper substantially.


Download Citation

Bernard Bercu. Manon Costa. Sébastien Gadat. "Stochastic approximation algorithms for superquantiles estimation." Electron. J. Probab. 26 1 - 29, 2021.


Received: 28 July 2020; Accepted: 22 May 2021; Published: 2021
First available in Project Euclid: 22 June 2021

Digital Object Identifier: 10.1214/21-EJP648

Primary: 62L20
Secondary: 60F05 , 62P05

Keywords: conditional value-at-risk , limit theorems , quantile and superquantile , stochastic approximation

Vol.26 • 2021
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