  The extreme statistics of time signals is studied when the maximum is
measured from the initial value. In the case of independent, identically
distributed (iid) variables, we classify the limiting distribution of the
maximum according to the properties of the parent distribution from which the
variables are drawn. Then we turn to correlated periodic Gaussian signals with
a 1/f^alpha power spectrum and study the distribution of the maximum relative
height with respect to the initial height (MRH_I). The exact MRH_I distribution
is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random
acceleration), 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 previously
studied distribution of the maximum height relative to the average height for
all alpha. The two main distinguishing features of the MRH_I distribution are
the much larger weight for small relative heights and the divergence at zero
height for alpha>3. We also demonstrate that the boundary conditions affect the
shape of the distribution by presenting exact results for some non-periodic
boundary conditions. Finally, we show that, for signals arising from
time-translationally invariant distributions, the density of near extreme
states is the same as the MRH_I distribution. This is used in developing a
scaling theory for the threshold singularities of the two distributions.
