Shapiro, Alexander Consistency of sample estimates of risk averse stochastic programs. (English) Zbl 1301.62045 J. Appl. Probab. 50, No. 2, 533-541 (2013). Summary: In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems. Cited in 8 Documents MSC: 62G20 Asymptotic properties of nonparametric inference 60F15 Strong limit theorems 90C15 Stochastic programming Keywords:law-invariant convex and coherent risk measures; stochastic programming; law of large numbers; consistency of statistical estimators; epiconvergence; sample average approximation PDFBibTeX XMLCite \textit{A. Shapiro}, J. Appl. Probab. 50, No. 2, 533--541 (2013; Zbl 1301.62045) Full Text: DOI Euclid References: [1] Artstein, Z. and Wets, R. J. B. (1996). Consistency of minimizers and the SLLN for stochastic programs. J. Convex Anal. 2 , 1-17. · Zbl 0837.90093 [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 , 203-228. · Zbl 0980.91042 · doi:10.1111/1467-9965.00068 [3] Billingsley, P. (1995). Probability and Measure , 3rd edn. John Wiley, New York. · Zbl 0822.60002 [4] Dupačová, J. and Wets, R. J. B. (1988). Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. 16 , 1517-1549. · Zbl 0667.62018 · doi:10.1214/aos/1176351052 [5] Föllmer, H. and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete Time , 2nd edn. Walter de Gruyter, Berlin. · Zbl 1126.91028 [6] Rockafellar, R. T. and Wets, R. J. B. (1998). Variational Analysis . Springer, Berlin. · Zbl 0888.49001 [7] Ruszczyński, A. and Shapiro, A. (2006). Optimization of convex risk functions. Math. Operat. Res. 31 , 433-452. · Zbl 1278.90283 · doi:10.1287/moor.1050.0186 [8] Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory . SIAM, Philadelphia. · Zbl 1183.90005 · doi:10.1137/1.9780898718751 [9] Wozabal, D. and Wozabal, N. (2009). Asymptotic consistency of risk functionals. J. Nonparametric Statist. 21 , 977-990. · Zbl 1175.62113 · doi:10.1080/10485250903060592 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.