On selfsimilar Lévy Random Probabilities

We explore a class of random probabilities induced by the normalization of selfsimilar Levy Random Measures–random measures whose probability laws are governed by stable one-sided Levy distributions. Various statistical properties of these random probabilities are analyzed: (i) moment structure; (ii) auto-covariance structure; (iii) one-dimensional and multidimensional tail-probabilities; and (iv) behavior in limiting cases. For the Levy–Smirnoff case—corresponding to the selfsimilarity index of order 12—an explicit analytic formula for the multidimensional probability density functions is derived. Last, a comparison between the class of selfsimilar Levy Random Probabilities and the Dirichlet Random Probability (induced by the normalization of the Gamma Random Measure) is conducted, showing the former to be far more robust than the latter.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[2]  C. Mallows,et al.  A Method for Simulating Stable Random Variables , 1976 .

[3]  B. Gnedenko,et al.  Limit Distributions for Sums of Independent Random Variables , 1955 .

[4]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[5]  P. Levy,et al.  Calcul des Probabilites , 1926, The Mathematical Gazette.

[6]  J. Bertoin.  Subordinators: Examples and Applications , 1999 .

[7]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[8]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[9]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[10]  M. Shlesinger,et al.  Beyond Brownian motion , 1996 .

[11]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[12]  I. Ibragimov,et al.  Independent and stationary sequences of random variables , 1971 .

[13]  V. Zolotarev,et al.  Chance and Stability, Stable Distributions and Their Applications , 1999 .

[14]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

[15]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[16]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[17]  On the extreme flights of one-sided Lévy processes , 2003, cond-mat/0307308.

[18]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[19]  P. Levy.  Processus stochastiques et mouvement brownien , 1948 .