Geodesic
On continuous convergence and epi-convergence of random functions. Part II: Sufficient conditions and applications
Kybernetika, Tome 39 (2003) no. 1, pp. 99-118 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Part II of the paper aims at providing conditions which may serve as a bridge between existing stability assertions and asymptotic results in probability theory and statistics. Special emphasis is put on functions that are expectations with respect to random probability measures. Discontinuous integrands are also taken into account. The results are illustrated applying them to functions that represent probabilities.
Part II of the paper aims at providing conditions which may serve as a bridge between existing stability assertions and asymptotic results in probability theory and statistics. Special emphasis is put on functions that are expectations with respect to random probability measures. Discontinuous integrands are also taken into account. The results are illustrated applying them to functions that represent probabilities.
Classification : 60B10, 62G05, 90C15, 90C31
Keywords: continuous convergence; epi-convergence; stochastic programming; stability; estimates 
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Vogel, Silvia; Lachout, Petr. On continuous convergence and epi-convergence of random functions. Part II: Sufficient conditions and applications. Kybernetika, Tome 39 (2003) no. 1, pp. 99-118. http://geodesic.mathdoc.fr/item/KYB_2003_39_1_a6/

[1] Artstein Z., Wets R. J.-B.: Stability results for stochastic programs and sensors, allowing for discontinuous objective functions. SIAM J. Optim. 4 (1994), 537–550 | DOI | MR | Zbl

[2] Bank B., Guddat J., Klatte D., Kummer, B., Tammer K.: Non-Linear Parametric Optimization. Akademie Verlag, Berlin 1982 | Zbl

[3] Billingsley P.: Convergence of Probability Measures. Wiley, New York 1968 | MR | Zbl

[4] Billingsley P.: Probability and Measure. Wiley, New York 1979 | MR | Zbl

[5] Devroye L., Györfi L.: Nonparametric Density Estimation. The L$_1$-View. Wiley, 1985 | MR | Zbl

[6] Dupačová J., Wets R. J.-B.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic problems. Ann. Statist. 16 (1988), 1517–1549 | DOI | MR

[7] Embrechts P., Klüppelberg, C., Mikosch T.: Modelling Extremal Events. Springer–Verlag, Berlin 1997 | MR | Zbl

[8] Györfi L., Härdle W., Sarda, P., Vieu P.: Nonparametric Curve Estimation from Time Series (Lecture Notes in Statistics 60). Springer–Verlag, Berlin 1989 | MR

[9] Kall P.: On approximations and stability in stochastic programming. In: Parametric Programming and Related Topics (J. Guddat, H. Th. Jongen, B. Kummer, and F. Nožička, eds.), Akademie Verlag, Berlin 1987, pp. 86–103 | MR | Zbl

[10] Kaniovski Y. M., King A. J., Wets R. J.-B.: Probabilistic bounds (via large deviations) for the solution of stochastic programming problems. Ann. Oper. Res. 56 (1995), 189–208 | DOI | MR

[11] Kaňková V.: A note on estimates in stochastic programming. J. Comput. Appl. Math. 56 (1994), 97–112 | DOI | MR

[12] Kaňková V., Lachout P.: Convergence rate of empirical estimates in stochastic programming. Informatica 3 (1992), 497–523 | MR | Zbl

[13] King A. J., Wets R. J.-B.: Epi-consistency of convex stochastic programs. Stochastics and Stochastics Reports 34(1991),83–92 | MR | Zbl

[14] Korf L. A., Wets R. J.-B.: Random lsc functions: an ergodic theorem. Math. Oper. Research 26 (2001), 421–445 | DOI | MR | Zbl

[15] Lachout P., Vogel S.: On continuous convergence and epi-convergence of random functions. Part I: Theory and relations. Kybernetika 39 (2003), 1, 75–98 | MR

[16] Langen H.-J.: Convergence of dynamic programming models. Math. Oper. Res. 6 (1981), 493–512 | DOI | MR | Zbl

[17] Liebscher E.: Strong convergence of sums of $\alpha $-mixing random variables with applications to density estimations. Stoch. Process. Appl. 65 (1996), 69–80 | DOI | MR

[18] Lucchetti R., Wets R. J.-B.: Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization. Statist. Decisions 11 (1993), 69–84 | MR | Zbl

[19] Pflug G. Ch., Ruszczyňski, A., Schultz R.: On the Glivenko–Cantelli problem in stochastic programming: Linear recourse and extensions. Math. Oper. Res. 23 (1998), 204–220 | DOI | MR | Zbl

[20] Rachev S. T.: The Monge–Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl. 29 (1984), 647–676 | MR

[21] Robinson S. M.: Local epi-continuity and local optimization. Math. Programming 37 (1987), 208–222 | DOI | MR | Zbl

[22] Robinson S. M., Wets R. J.-B.: Stability in two-stage stochastic programming. SIAM J. Control Optim. 25 (1987), 1409–1416 | DOI | MR | Zbl

[23] Römisch W., Schultz R.: Stability of solutions for stochastic programs with complete recourse. Math. Oper. Res. 18 (1993), 590–609 | DOI | MR | Zbl

[24] Silverman B. W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London 1986 | MR | Zbl

[25] Vogel S.: Stochastische Stabilitätskonzepte. Habilitation, Ilmenau Technical University, 1991

[26] Vogel S.: On stability in multiobjective programming – A stochastic approach. Math. Programming 56 (1992), 91–119 | DOI | MR | Zbl

[27] Vogel S.: A stochastic approach to stability in stochastic programming. J. Comput. Appl. Mathematics, Series Appl. Analysis and Stochastics 56 (1994), 65–96 | MR | Zbl

[28] Vogel S.: On stability in stochastic programming – Sufficient conditions for continuous convergence and epi-convergence. Preprint of Ilmenau Technical University, 1994 | MR

[29] Wang J.: Continuity of feasible solution sets of probabilistic constrained programs. J. Optim. Theory Appl. 63 (1989), 79–89 | DOI | MR

[30] Wets R. J.-B.: Stochastic programming. In: Handbooks in Operations Research and Management Science, Vol. 1, Optimization (G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, eds.), North Holland, Amsterdam 1989, pp. 573–629 | MR | Zbl

[31] Zervos M.: On the epiconvergence of stochastic optimization problems. Math. Oper. Res. 24 (1999), 2, 495–508 | DOI | MR | Zbl