Department of Mathematics

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

Alex Iksanov
(Kiev)

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