School of Mathematical Sciences

Asymptotics of the number of empty boxes in the Bernoulli sieve menu

Asymptotics of the number of empty boxes in the Bernoulli sieve

Speaker: 
Alexander Iksanov (National T. Shevchenko University, Kiev)
Date/Time: 
Wed, 20/03/2013 - 16:00
Room: 
M203

Suppose we are given a multiplicative random walk (a
stick-breaking set) generated by a random variable W taking values in the
interval (0,1) and a sample from the uniform [0,1] law which is
independent of the stick-breaking set. The Bernoulli sieve is a random
occupancy scheme in which 'balls' represented by the points of the uniform
sample are allocated over an infinite array of 'boxes' represented by the
gaps in the stick-breaking set. Assuming that the number of balls equals n
I am interested in the weak convergence of the number of empty boxes
within the occupancy range as n approaches infinity. Depending on the
behavior of the law of W near the endpoints 0 and 1 the number of empty
boxes can exhibit quite a wide range of different asymptotics. I will
discuss the most interesting cases with an emphasis on the methods
exploited.