Abstract: |
We consider the occurrence of record-breaking events in random walks with
asymmetric jump distributions. The statistics of records in symmetric random
walks was previously analyzed by Majumdar and Ziff and is well understood.
Unlike the case of symmetric jump distributions, in the asymmetric case the
statistics of records depends on the choice of the jump distribution. We
compute the record rate $P_n(c)$, defined as the probability for the $n$th
value to be larger than all previous values, for a Gaussian jump distribution
with standard deviation $\sigma$ that is shifted by a constant drift $c$. For
small drift, in the sense of $c/\sigma \ll n^{-1/2}$, the correction to
$P_n(c)$ grows proportional to arctan$(\sqrt{n})$ and saturates at the value
$\frac{c}{\sqrt{2} \sigma}$. For large $n$ the record rate approaches a
constant, which is approximately given by
$1-(\sigma/\sqrt{2\pi}c)\textrm{exp}(-c^2/2\sigma^2)$ for $c/\sigma \gg 1$.
These asymptotic results carry over to other continuous jump distributions
with finite variance. As an application, we compare our analytical results to
the record statistics of 366 daily stock prices from the Standard & Poors 500
index. The biased random walk accounts quantitatively for the increase in the
number of upper records due to the overall trend in the stock prices, and
after detrending the number of upper records is in good agreement with the
symmetric random walk. However the number of lower records in the detrended
data is significantly reduced by a mechanism that remains to be identified. |