[JEL: C12, C15, C40]

*Atlantic Economic Journal 28:4, December 2000*

**ROBERT F. MULLIGAN**

Western Carolina University and the State University of New York at
Binghamton

The Mandelbrot-Levy distributions are a family of infinite-variance
distributions without explicit analytical expressions, except for special
cases. Limiting distributions include the normal, with finite variance,
and the Cauchy, with the most extreme platykurtosis or fat tails. Paul
Levy developed the theory of these distributions [*Calcul des Probibilites*,
Paris: Gauthier Villars, 1925]. The Hurst exponent H introduced in
the hydrological study of the Nile valley is the reciprocal of the characteristic
exponent [*Transactions of the American Society of Civil Engineers*,
1951, v. 116, pp. 770-799].

The characteristic function of a Mandelbrot-Levy random variable is:

Mandelbrot introduced a technique for estimating alpha by regression
[*Annals of Economic and Social Measurement*, 1972, 1:3, pp. 259-290],
further refined by Lo [*Econometrica*, 1991, 59:3, pp. 1279-1313].
Various statistics are available to test the null hypothesis of normality,
but not for the Cauchy distribution, the other extreme.

Tables of percentages of (alpha) were generated by Monte Carlo experiments with 1,000 iterations for different sample sizes. Here follow the sample sizes, followed by 1%, 5%, 10%, 90%, 95% and 99% critical values:

[N=50:
[N=100:
[N=250:
[N=500:
[N=1,000:
[N=2,500:
[N=5,000:
[N=10,000: |
0.254; 0.464; 0.550; 1.330;
1.521; 1.965],
0.410; 0.556; 0.631; 1.256; 1.354; 1.634],
0.526; 0.691; 0.749; 1.157; 1.215; 1.324],
0.659; 0.765; 0.802; 1.108; 1.153; 1.227],
0.699; 0.825; 0.854; 1.085; 1.108; 1.154],
0.807; 0.876; 0.899; 1.045; 1.066; 1.096],
0.864; 0.906; 0.926; 1.032; 1.048; 1.071],
0.885; 0.935; 0.946; 1.023; 1.034; 1.049]. |