Multifractional Probabilistic Laws
- 2.3k Downloads
In this paper, we apply the theory of pseudodifferential operators and Sobolev spaces to characterize fractional and multifractional probability densities. In the fractional case, local regularity properties of the probability density function are given in terms of fractional moment conditions satisfied by the characteristic function. Conversely, the parameter defining the order of the fractional Sobolev space where the characteristic function lies provides the index of stability in relation to fractional moment conditions of the probability density. The extension to the multifractional case leads to the introduction of new probabilistic models considering the theory of pseudodifferential operators and fractional Sobolev spaces of variables order.
Keywords and pharasesBessel distribution fractional pseudodifferential operators Laplace distribution multifractional pseudodifferential operators
Unable to display preview. Download preview PDF.
- 1.Dautray, R., and Lions, J. L. (1985). Mathematical Analysis and Numerical Methods for Science and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, New York.Google Scholar
- 8.Kemp, F. (2003). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Journal of the Royal Statistical Society, Series D, 52, 698–699.Google Scholar
- 9.Kikuchi, K., and Negoro, A. (1995). Pseudodifferential operators with variable order of differentiation. Reports of Liberal Arts and Science Faculty, Shizuoka University, 31, 19–27.Google Scholar
- 15.Stein, E. M. (1970). Singular Integrals and Differential Properties of Functions, Princeton University Press, Princeton, NJ.Google Scholar
- 16.Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co., Amsterdam.Google Scholar