Event Detail

Event Type: 
Probability Seminar
Tuesday, January 26, 2016 - 16:00 to 17:00
WNGR 201

Speaker Info

University of Sao Paulo, Brazil

A renewal process counts the number of "events" in the case where the gaps between successive events are independent of each other and are distributed in the same way. One example is the number of burnt-out light bulbs up to a certain time; another is the number of returns to a given state of a countable state Markov process, for example the returns to zero of a random walk. Given the gap distribution, one can calculate the expected return time; there is a dichotomy between this (the first moment) being finite or infinite. One can make a finer distinction, considering the least moment alpha that is finite. Of particular importance is the second moment (the variance); whether or not this is finite subdivides the finite expectation region in two. Making the assumption of regularly varying tails, the result is three "phases" of asymptotic behavior: Gaussian, stable and Mittag-Leffler.
From the point of ergodic theory, the renewal process counts the number of returns to a subset of finite measure, and as we transition through the three phases, we can observe the transition from finite to infinite invariant measure. Precisely, we show that in each case one has asymptotically self-similar returns, stated as an almost-sure invariance principle in log density. For the infinite measure case this is interpreted as fractal-like return structure, leading to an order-two ergodic theorem.

Joint work with Marina Talet.