Probability Distribution of the Maximum of a Smooth Temporal Signal

Abstract

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated to the probability that the process remains below a non-zero level M . When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f (t) = X(0)X(t) .The problem of evaluating the distribution of the maximum of a time-correlated random variable X(t) has elicited a large body of work by mathematicians [1,2,3], and physicists, both theorists [4,5,6,7,8,9,10,11,12] and experimentalists [13,14,15,16,17]. In the physics literature, this is related to the persistence problem, the probability that a temporal signal X (and hence its maximum) remains below a given level M up to time t. The mathematical literature has mainly focused on evaluatingfor Gaussian processes and for large |M |, a regime where efficient bounds or equivalent have been obtained [1,2]. Recently [3], and for Gaussian processes only, a numerical method to obtain valuable bounds has been extended to all values of M , although the required numerical effort can become quite considerable for large t.Physicists have also concentrated their attention to Gaussian processes [7,8,9,10], which are often a good or exact description of actual physical processes. For instance, the total magnetization in a spin system [11], or the height profile of certain fluctuating interfaces [12,15,16] are true temporal Gaussian processes. Two general methods have been developed, focusing on the case M = 0, which applies to many physical situations. The first one [7,8,9] is a perturbation of the considered process around the Markovian Gaussian process, which has been extended for small values of M [8]. Within this method, only the large time asymptotics of P < (t) is known, leading to the definition of the persistence exponent (see below). The alternative method, using the independent interval approximation [10], gives very accurate results for smooth processes, but is restricted to M = 0.In addition, this problem has obvious applications in many other applied and experimental sciences, where one has to deal with data analysis of complex statistical signals. For instance, statistical bounds of noisy signals are extremely useful for image processing (for instance in medical imaging or astrophysics [18]), in order to obtain cleaner images by correcting spurious bright or dark pixels [1,3]. In general, it is important to be able to evaluate the maximum of a correlated temporal or spatial signal originating from experimental noise. The same question can arise when the signal lives in a more abstract space. For instance, in the context of genetic cartography, statistical methods to evaluate the maximum of a complex signal has been exploited to identify putative quantitative trait loci [19]. Finally, this same problem arises in econophysics or finance, where the probability for a generally strongly correlated financial signal to remain below or ab...

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