Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

Suppose a discrete-time signal S(t), 0/spl les/t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1/sub {t=/spl tau/}/ and sinusoids exp{2/spl pi/iwt/N}//spl radic/N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l/sup 1/ norm of the coefficients among all decompositions. Here "highly sparse" means that N/sub t/+N/sub w/</spl radic/N/2 where N/sub t/ is the number of time atoms, N/sub w/ is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l/sup 1/ uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l/sup 1/ norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2/sup j/2/.

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Uncertainty principles and ideal atomic decomposition

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Conditions for a process ζ on a compact metric spaceS to be simultaneously max-infinitely divisible and sample continuous are obtained. Although they fall short of a complete characterization of such processes, these conditions yield complete descriptions of the sample continuous non-degenerate max-stable processes onS and of the infinitely divisible non-void random compact subsets of a Banach space under the operation of convex hull of unions.

Max-infinitely divisible and max-stable sample continuous processes

Limit theorems for random sets: An application of probability in banach space results

In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is the first hitting time process for either a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property.

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Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion

It is known that the transition probabilities of a solution to a classical Ito stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Levy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.

SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations

Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {Xn} of independent copies of X, let Sn=X1+⋯+Xn. Conditions which are either necessary or sufficient for the weak convergence of n−1/2(Sn−ESn) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.

Marjorie G. Hahn

Papers

Max-infinitely divisible and max-stable sample continuous processes

Limit theorems for random sets: An application of probability in banach space results

Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion

SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations

Central limit theorems in D[0, 1]