3:00 pm Wednesday, November 4, 2015
Random Structures: Mass transports between stationary random measures by
Ali Khezeli (University of Texas at Austin) in RLM 8.136
The problem of finding a transport kernel between (samples of) stationary random measures was initiated by [Thorisson, H. (1996). Transforming random elements and shifting random fields. The Annals of Probability, 24(4), 2057-2064.] with a necessary and sufficient condition for two random measures to be obtained by a random translation from each other, which is called a shift coupling. A particular example is when the second ranom measure is the Palm version of the first one. In the case of point processes a number of algorithms are presented to construct a shift coupling, which we will review. In particular, we are interested in the algorithm presented in [Hoffman, C., Holroyd, A. E., & Peres, Y. (2006). A stable marriage of Poisson and Lebesgue. The Annals of Probability, 1241-1272], which extends the Gale-Shapley stable marriage algorithm and the notion of stability to a continuum setting. We then generalize it for arbitrary random measures, which is the work of my Ph.D. thesis. For this, we limit ourselves to 'mild' transport kernels, which is a special case of capacity constrained transport kernels. We give a definition of stability of mild transport kernels and introduce a construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable mild transport kernels, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness. Submitted by
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