Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes

4:20 – 5:00

Wlodzimierz Bryc (U Cincinnati)
Martingales from pairs of randomized Poisson, Gamma,
negative binomial and hyperbolic secant processes

Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes
Wlodzimierz Bryc (U Cincinnati)
Abstract: Consider a pair of independent Poisson processes, or a pair of Negative Binomial processes, or Gamma, or hyperbolic secant processes with a shared randomly selected parameter. Under appropriate randomization, one can deterministically re-parametrize the time and scale for both processes so that the first process runs on time interval (0,1), the second process runs on time interval (1,∞), and the two processes seamlessly join into one Markov martingale on (0,∞). In fact, a property stronger than martingale holds: we stitch together two processes into a single quadratic harness on (0,∞). This talk is based on joint work in progress with J. Wesolowski.