Let two moving averages y 1 and y 2 be calculated, respectively, over e.g. T1 and T2
intervals such that T2>T1. If y(t) increases for a long period before decreasing rapidly,
y 1 will cross y 2 from above. In empirical nance this event is called a death cross
because the signal measured on a short-time interval decreases faster than the overall
trend, as measure by the average in the longer interval [23]. This leads to pessimism
concerning the behavior of the signal which should later hit some minimum. On the
contrary, if y 1 crosses y 2 from below, the crossing point coincides with an upsurge of
the signal y(t), { such a crossing is called a gold cross by optimism; it occurs before
a maximum. The density of crossing points between any two moving averages is
obviously a measure of long-range power-law correlations in the signal.
However the density of crossing points ( T), where T = (T2 − T1)=T2, is fully
symmetric, has a minimum [24] in the middle of the T interval and diverges for
T = 0 and for T = 1, with an exponent which is the Hurst exponent. This result
certainly raises fundamental questions on the properties of (fractional or not) Brownian
motion processes. The behavior of is analogous to the density of electronic states on a
fractal lattice in a tight-binding approximation [25,26]. Notice that the moving-average
method can serve to measure the Hurst exponent in a very fast, elegant and continuous
way.
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