| Title:
|
On the Argmin-sets of stochastic processes and their distributional convergence in Fell-type-topologies (English) |
| Author:
|
Ferger, Dietmar |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
47 |
| Issue:
|
6 |
| Year:
|
2011 |
| Pages:
|
955-968 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\epsilon-\text(Z)$ be the collection of all $\epsilon$-optimal solutions for a stochastic process $Z$ with locally bounded trajectories defined on a topological space. For sequences $(Z_n)$ of such stochastic processes and $(\epsilon_n)$ of nonnegative random variables we give sufficient conditions for the (closed) random sets $\epsilon_n-\text(Z_n)$ to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology. (English) |
| Keyword:
|
$\epsilon$-argmin of stochastic process |
| Keyword:
|
random closed sets |
| Keyword:
|
weak convergence of Hoffmann--Jørgensen |
| Keyword:
|
Fell-topology |
| Keyword:
|
Missing-topology |
| MSC:
|
49J53 |
| MSC:
|
60B10 |
| MSC:
|
60F05 |
| MSC:
|
90C15 |
| idZBL:
|
Zbl 1241.93054 |
| idMR:
|
MR2907854 |
| . |
| Date available:
|
2011-12-08T10:07:09Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141737 |
| . |
| Reference:
|
[1] Billingsley, P.: Convergence of Probability Measures.John Wiley & Sons, New York 1968. Zbl 0172.21201, MR 0233396 |
| Reference:
|
[2] Ferger, D.: A continuous mapping theorem for the Argmax-functional in the non-unique case.Statist. Neerlandica 58 (2004), 83–96. Zbl 1090.60032, MR 2042258, 10.1046/j.0039-0402.2003.00111.x |
| Reference:
|
[3] Gänssler, P., Stute, W.: Wahrscheinlichkeitstheorie.Springer–Verlag, Berlin – Heidelberg 1977. MR 0501219 |
| Reference:
|
[4] Gersch, O.: Convergence in Distribution of Random Closed Sets and Applications in Stability Theory and Stochastic Optimization.PhD Thesis. Technische Universität Ilmenau 2007. |
| Reference:
|
[5] Kallenberg, O.: Foundations of Modern Probability.Springer–Verlag, New York 1997. Zbl 0892.60001, MR 1464694 |
| Reference:
|
[6] Lagodowski, A., Rychlik, Z.: Weak convergence of probability measures on the function space $D_d[0,\infty )$.Bull. Polish Acad. Sci. Math. 34 (1986), 329–335. MR 0874876 |
| Reference:
|
[7] Lindvall, T.: Weak convergence of probability measures and random functions in the function space $D[0,\infty )$.J. Appl. Probab. 10 (1973), 109–121. Zbl 0258.60008, MR 0362429, 10.2307/3212499 |
| Reference:
|
[8] Norberg, T.: Convergence and existence of random set distributions.Ann. Probab. 12 (1984), 726–732. Zbl 0545.60021, MR 0744229, 10.1214/aop/1176993223 |
| Reference:
|
[9] Pflug, G. Ch.: Asymptotic dominance and confidence for solutions of stochastic programs.Czechoslovak J. Oper. Res. 1 (1992), 21–30. Zbl 1015.90511 |
| Reference:
|
[10] Pflug, G. Ch.: Asymptotic stochastic orograms.Math. Oper. Res. 20 (1995), 769–789. MR 1378105, 10.1287/moor.20.4.769 |
| Reference:
|
[11] Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis.Springer–Verlag, Berlin – Heidelberg 1998. Zbl 0888.49001, MR 1491362 |
| Reference:
|
[12] Royden, H. L.: Real Analysis.Third edition Macmillan Publishing Company, New York 1988. Zbl 0704.26006, MR 1013117 |
| Reference:
|
[13] Salinetti, G., Wets, R. J.-B.: On the convergence in distribution of measurable multifunctions (random sets), normal integrands, stochastic processes and stochastic infima.Math. Oper. Res. 11 (1986), 385–419. Zbl 0611.60004, MR 0852332, 10.1287/moor.11.3.385 |
| Reference:
|
[14] Vaart, A. W. van der, Wellner, J. A.: Weak Convergence and Empirical Processes.Springer–Verlag, New York 1996. MR 1385671 |
| Reference:
|
[15] Vogel, S.: Qualitative stability of stochastic programs with applications in asymptotic statistics.Statist. Decisions 23 (2005), 219–248. Zbl 1093.62032, MR 2236458, 10.1524/stnd.2005.23.3.219 |
| Reference:
|
[16] Vogel, S.: Semiconvergence in distribution of random closed sets with applications to random optimization problems.Ann. Oper. Res. 142 (2006), 269–282. MR 2222921, 10.1007/s10479-006-6172-0 |
| . |
