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The extreme statistics of time signals is studied when the maximum ismeasured from the initial value. In the case of independent, identicallydistributed (iid) variables, we classify the limiting distribution of themaximum according to the properties of the parent distribution from which thevariables are drawn. Then we turn to correlated periodic Gaussian signals witha 1/f^alpha power spectrum and study the distribution of the maximum relativeheight with respect to the initial height (MRH_I). The exact MRH_I distributionis derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (randomacceleration), and alpha=infinity (single sinusoidal mode). For other,intermediate values of alpha, the distribution is determined from simulations.We find that the MRH_I distribution is markedly different from the previouslystudied distribution of the maximum height relative to the average height forall alpha. The two main distinguishing features of the MRH_I distribution arethe much larger weight for small relative heights and the divergence at zeroheight for alpha>3. We also demonstrate that the boundary conditions affect theshape of the distribution by presenting exact results for some non-periodicboundary conditions. Finally, we show that, for signals arising fromtime-translationally invariant distributions, the density of near extremestates is the same as the MRH_I distribution. This is used in developing ascaling theory for the threshold singularities of the two distributions.

Extreme statistics for time series: Distribution of the maximum relative to the initial value