Universiteit Utrecht

Department of Mathematics

Variations on Ignatov's theorem

Sasha Gnedin (UU)
11 September 2007

Ignatov's theorem is a fundamental result of probability theory asserting that the point processes of $k$-record values from a sample $X_1,X_2,\ldots$ are iid. Observation $X_n$ is said to be a $k$-record if it is the $k$th smallest at time $n$. In the talk this result will be first reviewed from the prospective of a planar Poisson process, which exhibits symmetry between record values and record times, and also offers a path to a remarkable scale-invariant spacings lemma due to Arratia, Barbour and Tavare. A novel element is that we focus on the planar process of $k$-corners, a kind of an envelope for the processes of $i$-records for $i\leq k$. This covers the instance of continuously distributed $X_n$'s. Then a combinatorial approach will be sketched, leading to Ignatov's theorem in its most general form, with arbitrary distribution for the $X_n$'s.

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Charlene Kalle