# Article

 Title: A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes (English) Author: Ferger, Dietmar Language: English Journal: Kybernetika ISSN: 0023-5954 (print) ISSN: 1805-949X (online) Volume: 57 Issue: 3 Year: 2021 Pages: 426-445 Summary lang: English . Category: math . Summary: For lower-semicontinuous and convex stochastic processes $Z_n$ and nonnegative random variables $\epsilon_n$ we investigate the pertaining random sets $A(Z_n,\epsilon_n)$ of all $\epsilon_n$-approximating minimizers of $Z_n$. It is shown that, if the finite dimensional distributions of the $Z_n$ converge to some $Z$ and if the $\epsilon_n$ converge in probability to some constant $c$, then the $A(Z_n,\epsilon_n)$ converge in distribution to $A(Z,c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in contrast to these argmin-theorems we do not require that the limit process has a unique minimizing point. In the non-unique case the limit-distribution is replaced by a Choquet-capacity. (English) Keyword: convex stochastic processes Keyword: sets of approximating minimizers Keyword: weak convergence Keyword: Vietoris hyperspace topologies Keyword: Choquet-capacity MSC: 60B05 MSC: 60B10 MSC: 60F99 idZBL: Zbl 07442518 idMR: MR4299457 DOI: 10.14736/kyb-2021-3-0426 . Date available: 2021-11-04T12:43:53Z Last updated: 2022-02-24 Stable URL: http://hdl.handle.net/10338.dmlcz/149200 . Reference: [1] Attouch, H.: Variational Convergence for Functions and Operators..Applicable Mathematics Series, Pitmann, London 1984. Reference: [2] Bertsekas, D. P.: Convex Analysis and Optimization..Athena Scientific, Belmont, Massachusetts 2003. Reference: [3] Chernozhukov, V.: Extremal quantile regression..Ann. Statist. 33 (2005), 806-839. Reference: [4] Davis, R. A., Knight, K., Liu, J.: M-estimation for autoregressions with infinite variance..Stochastic Process. Appl. 40 (1992), 145-180. Reference: [5] Ferger, D.: Weak convergence of probability measures to Choquet capacity functionals..Turkish J. Math. 42 (2018), 1747-1764. Reference: [6] Gaenssler, P., Stute, W.: Wahrscheinlichkeitstheorie..Springer-Verlag, Berlin, Heidelberg, New York 1977. Zbl 0357.60001 Reference: [7] Geyer, C. J.: On the asymptotics of convex stochastic optimization..Unpublished manuscript (1996). Reference: [8] Haberman, S. J.: Concavity and estimation..Ann. Statist. 17 (1989), 1631-1661. Reference: [9] Hjort, N. L., Pollard, D.: Asymptotic for minimizers of convex processes..Preprint, Dept. of Statistics, Yale University (1993). Reference: [10] Hoffmann-Jørgensen, J.: Convergence in law of random elements and random sets..In: High Dimensional Probability (E. Eberlein, M. Hahn and M. Talagrand, eds.), Birkhäuser Verlag, Basel 1998, pp. 151-189. Reference: [11] Kallenberg, O.: Foundations of Modern Probability..Springer-Verlag, New York 1997. Zbl 0996.60001 Reference: [12] Knight, K.: Limiting distributions for $L_1$ regression estimators under general conditions..Ann. Statist. 26 (1998), 755-770. Reference: [13] Knight, K.: Limiting distributions of linear programming estimators..Extremes 4 (2001), 87-103. Reference: [14] Knight, K.: What are the limiting distributions of quantile estimators?.In: Statistical Data Analysis Based on the $L_1$-Norm and Related Methods (Y. Dodge, ed.), Series Statistics for Industry and Technology, Birkhäuser Verlag, Basel pp. 47-65. Reference: [15] Liese, F., Mieschke, K-J.: Statistical Decision Theory..Springer Science and Business Media, LLC, New York 2008. Reference: [16] Molchanov, I.: Theory of Random Sets. Second Edition..Springer-Verlag, New York 2017. Reference: [17] Pflug, G. Ch.: Asymptotic dominance and confidence regions for solutions of stochastic programs..Czechoslovak J. Oper. Res. 1 (1992), 21-30. Reference: [18] Pflug, G. Ch.: Asymptotic stochastic programs..Math. Oper. Res. 20 (1995), 769-789. Reference: [19] Rockefellar, R. T., Wets, R. J.-B.: Variational Analysis..Springer-Verlag, Berlin, Heidelberg 1998. Reference: [20] Smirnov, N. V.: Limiting distributions for the terms of a variational series..Amer. Math. Soc. Trans. 67 (1952), 82-143. Reference: [21] Topsøe, F.: Topology and Measure. Lecture Notes in Mathematics..Springer-Verlag, Berlin, Heidelberg, New York 1970. Reference: [22] Wagener, J., Dette, H.: Bridge estimators and the adaptive Lasso under heteroscedasticity..Math. Methods Statist. 21 (2012), 109-126. .

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