Kybernetika 47 no. 6, 955-968, 2011

On the argmin-sets of stochastic processes and their distributional convergence in Fell-type-topologies

Dietmar Ferger

Abstract:

Let <span class="tex">&epsilon;</span>-Argmin<span class="tex">(Z)</span> be the collection of all <span class="tex">&epsilon;</span>-optimal solutions for a stochastic process <span class="tex">Z</span> with locally bounded trajectories defined on a topological space. For sequences <span class="tex">(Z<sub>n</sub>)</span> of such stochastic processes and <span class="tex">(&epsilon;<sub>n</sub>)</span> of nonnegative random variables we give sufficient conditions for the (closed) random sets <span class="tex">&epsilon;<sub>n</sub></span>-Argmin<span class="tex">(Z<sub>n</sub>)</span> to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.

Keywords:

$\epsilon -$argmin of stochastic process, random closed sets, weak convergence of Hoffmann-Jørgensen, Fell-topology, Missing-topology

Classification:

49J53, 60B10, 60F05, 90C15

paper.pdf

References:

  1. P. Billingsley: Convergence of Probability Measures. John Wiley \et Sons, New York 1968.   CrossRef
  2. D. Ferger: A continuous mapping theorem for the Argmax-functional in the non-unique case. Statist. Neerlandica 58 (2004), 83-96.   CrossRef
  3. P. Gänssler and W. Stute: Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin - Heidelberg 1977.   CrossRef
  4. O. Gersch: Convergence in Distribution of Random Closed Sets and Applications in Stability Theory and Stochastic Optimization. PhD Thesis. Technische Universität Ilmenau 2007.   CrossRef
  5. O. Kallenberg: Foundations of Modern Probability. Springer-Verlag, New York 1997.   CrossRef
  6. A. Lagodowski and Z. Rychlik: Weak convergence of probability measures on the function space $D_d[0,\infty)$. Bull. Polish Acad. Sci. Math. 34 (1986), 329-335.   CrossRef
  7. T. Lindvall: Weak convergence of probability measures and random functions in the function space $D[0,\infty)$. J. Appl. Probab. 10 (1973), 109-121.   CrossRef
  8. T. Norberg: Convergence and existence of random set distributions. Ann. Probab. 12 (1984), 726-732.   CrossRef
  9. G. Ch. Pflug: Asymptotic dominance and confidence for solutions of stochastic programs. Czechoslovak J. Oper. Res. 1 (1992), 21-30.   CrossRef
  10. G. Ch. Pflug: Asymptotic stochastic orograms. Math. Oper. Res. 20 (1995), 769-789.   CrossRef
  11. R. T. Rockafellar and R. J.-B. Wets: Variational Analysis. Springer-Verlag, Berlin - Heidelberg 1998.   CrossRef
  12. H. L. Royden: Real Analysis. Third edition Macmillan Publishing Company, New York 1988.   CrossRef
  13. G. Salinetti and R. J.-B. Wets: On the convergence in distribution of measurable multifunctions (random sets), normal integrands, stochastic processes and stochastic infima. Math. Oper. Res. 11 (1986), 385-419.   CrossRef
  14. A. W. van der Vaart and J. A. Wellner: Weak Convergence and Empirical Processes. Springer-Verlag, New York 1996.   CrossRef
  15. S. Vogel: Qualitative stability of stochastic programs with applications in asymptotic statistics. Statist. Decisions 23 (2005), 219-248.   CrossRef
  16. S. Vogel: Semiconvergence in distribution of random closed sets with applications to random optimization problems. Ann. Oper. Res. 142 (2006), 269-282.   CrossRef