Gregor Wergen

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I will discuss the statistics of the number of records for N identical
and independent symmetric discrete-time random walks. At each time step,
each walker jumps by a random length drawn independently from a
symmetric and continuous distribution. One has to distinguish between
two cases: (I) when the variance of the jump distribution is finite and
(II) when it is divergent as in the case of Levy flights. In both
cases the mean record number grows universally with the square root of n
for large n, but with a very different behavior of the amplitude in the
two cases. For a finite variance I will argue, and this is confirmed by
numerical simulations, that the full distribution of the record number
converges to a Gumbel law for large n. In case II, numerical simulations
indicate that the distribution of the record number converges, for large
n and N, to a universal nontrivial distribution, independently of the
Levy index. I also discuss an application of these results to the
study of the record statistics of daily stock prices.

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Institute for Theoretical Physics, Cologne
Konferenzraum Theoretische Physik
Contact: not specified