Universiteit Utrecht

Department of Mathematics

The Bernoulli sieve revisited (based on joint work with A.Gnedin (Utrecht), P.Negadajlov (Kiev) and U.Roesler (Kiel))

Alex Iksanov

12 February 2008

The talk is concerned with a "balls-in-boxes" scheme with random frequencies where "balls" are identified with a random sample of size $n$ from the exponential distribution with unit mean, and the role of "boxes" play intervals whose endpoints are successive points of a random walk with positive and non-lattice steps. I will discuss the weak limiting behaviour, as $n\to\infty$, of the five functionals arising from the scheme: $U_n$--the number (in order) of the last occupied interval; $K_n$--the number of occupied intervals; $K_{n,\,0}$--the number of empty intervals among the first $U_n-1$; $W_n$-- the number (in order) of the first empty interval; $Z_n$--the number of points in the last occupied interval. It will be explained that the weak asymptotic behaviour of $U_n$ coincides with that of the first passage time through the level $\log n$ by the random walk. It turns out that the same is true for $K_n$ and $W_n$ under a side condition that prevents occurrence of very short intervals. Among other things, this condition ensures a rather delicate result: $K_{n,0}$ weakly converges (without normalization). The talk closes with stating limiting results for $Z_n$ under (1) finite mean; (2) regular variation assumptions.

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Charlene Kalle