Multifractality of US Dollar/Deutsche Mark Exchange Rates - Studies2

• α0 denotes the value of α for which f(α0) = 1. This is the maximum value of the multifractal
spectrum. By strict concavity, α0 is unique. This is also the most frequently occurring value
of α, occupying a set of Lebesgue measure one (or T if the sample span is not renormalized
to a value of one.)
• α1 denotes the value of α that carries most of the mass of a multifractal measure, or most
of the variation of a multifractal process. Multifractal measures satisfy f(α1) =α1. The
spectrum of a multifractal process generated by the MMAR thus satisfies fP (α1) =1/H.
• αmin denotes the smallest manifest α, i.e. αmin =
• αmax denotes the largest manifest α.
min : f(α) ≥ 0 .
α
Each of these points is depicted in the multifractal diagram shown in Figure 8.
4.4 Unifractal Examples
While the Legendre transform (8) can be worked out mechanically, a graphical representation is
more instructive. This is obtained by plotting, for each q, a line of slope q and vertical intercept
−τ(q). Placing all of these lines upon the same graph, the lower envelope gives the estimated
multifractal spectrum.
For each of the Fractional Gaussian Noises in Figure 4, the simulated partition functions follow
lines of approximate slope τ(q) =Hq − 1. Figure 9 shows Legendre transforms of scaling functions
for FGN’s with self–affinity indices .4, .53, .6, .8. Only the left side of each spectrum is shown.
The right side of the multifractal spectrum comes from negative moments of the partition function,
which are not used in the empirical work in this paper. Each of the lines drawn passes through the
same point at {α, f(α)} = {H, 1} . The lower envelope is thus degenerate:
⎧
⎪⎨ 1, α = H
f(α) =
⎪⎩ −∞, α = H
By inspection of equation (8), this form derives from the linearity of τ(q). The multifractal spectrum
is defined only for values of α that are tangency slopes for some value of the scaling function.
A single valued multifractal spectrum is an alternative way of saying that each of these processes
contains only a single Hölder exponent. Thus α0, which carries all of the Lebesgue measure of the
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