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- 20 octobre 2006
One is often interested in predicting the maximum or minimum in a given time interval of a random but correlated temporal signal .
Defining the probability that the process (and hence its maximum) remains below the value (), its knowledge provides valuable information on the underlying physical system associated to . For large time, one observes in many physical system, or for stationary systems, thus defining the persistence exponent . This general problem has applications in
Previous analytical results for and have addressed the case (when is of zero average) and have focused on the simpler case where is a Gaussian process. In a recent article, a member of the laboratory has obtained an explicit form for from a minimal knowledge of the properties of a general process (for instance, its correlation function ). In addition, the distributions of time intervals when the signal remains below or above the level and the persistence exponent have been computed. The calculation relies on the approximation that the lengths of the intervals between successive crossings of the level are uncorrelated, an assumption becoming exact for large .
The corresponding paper Probability distribution of the maximum of a smooth temporal signal has been published in Physical Review Letters (2007), and is available on cond-mat.
The corresponding author in the laboratory is Clément Sire.
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