Conditional multidimensional random walks: Construction and limiting behaviour

• Applicant: Professor Dr. Vitali Wachtel
• Subject Area: Mathematics
• Term: from 2010 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 188691779

• Final Report Year: 2014

Final Report Abstract

For a long time exit times for random walks have been studied only via duality relations and corresponding factorisation identities. The main achievment of our project is a completely new approach to the study of exit times and corresponding conditioned distributions of Markov processes in discrete time. This method does not use duality, is quite robust and allows to obtain results for various models: multidimensional random walks in cones, integrated random walks and Markov chains with asymptotically zero drift.

Publications

• Random walks in cones. Ann. Probab.
  D. Denisov and V. Wachtel
  
• Exit times for integrated random walks. Ann. Inst. H. Poincare Probab. Statist.
  D. Denisov and V. Wachtel
  
• Martingale approach to subexponential asymptotics for random walks. Electron. Commun. Probab., 17(2012), Paper no, 6, 1-9
  D. Denisov and V. Wachtel
  (See online at https://doi.org/10.1214/ECP.v17-1757)
• Ordered random walks with heavy tails. Electron. J. Probab., 17(2012), Paper no, 5, 1-21
  D. Denisov and V. Wachtel
  
• Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics. Stochastic Process. Appl., 123(2013): 3027-3051
  D. Denisov, D. Korshunov and V. Wachtel
  (See online at https://doi.org/10.1016/j.spa.2013.04.011)
• Upper bounds for the maximum of a random walk with negative drift. J. Appl. Probab., 50(2013): 1131-1046
  J. Kugler and V. Wachtel
  
• Heavy traffic and heavy tails for subexponential distributions
  D. Denisov and J.Kugle
  
• Local limit theorem for the maximum of a random walk
  J. Kugler