William N. Hudson

Operator-self-similar processes in a finite-dimensional space

A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.

Operator-stable laws

Sharpe investigated the structure of full operator-stable measures [mu] on a vector group V and obtained decompositions, [mu] = [mu]1 * [mu]2 and V = V1 [circle plus operator] V2, in terms of the Gaussian component [mu]1 and the Poisson component [mu]2. The subspaces V1 and V2 are here identified in terms of an exponent B for [mu]. Sharpe also pointed out that the Levy measure M of [mu] is a mixture of Levy measures concentrated on single orbits of tB. Here, an explicit representation is obtained for M as such a mixture by constructing a measure on the unit sphere. Also, necessary and sufficient conditions are given that a Levy measure be the Levy measure of a full operator-stable measure. The final result deals with full Gaussian measures [mu] and establishes the connection between its covariance operator and the class of all exponents of [mu].

Variational sums for additive processes

Let X(t), 0 c t c T, be an additive process, and let Xnk be the kth increment of X(t) associated with the partition rIn of [0, T]. Assume IIR, II -n| 0. Let f3 be the Blumenthal-Getoor index of X(T) and let 2 _ y > /8. When the partitions are nested, k IXkly converges a.s. to I {IJ(s)IT: 0 _ s _ T}, where J(s) is the jump of X(t) at s. This convergence also holds when the partitions are not nested provided either X(t) has stationary increments or 1 _ y > ,B. This extends a result of P. W. Millar and completes a result of S. M. Berman.

Moments of distributions attracted to operator-stable laws

Bounds on the norming operators for distributions in the domain of attraction of an operator-stable distribution are found. These bounds are used to establish the existence and nonexistence of moments of distributions in the domain of attraction of an operator-stable distribution. Similar results for stochastically compact sequences are obtained.

Operator-stable distributions and stable marginals

Sharpe has shown that full operator-stable distributions [mu] on Rn are infinitely divisible and for a suitable automorphism B depending on [mu] satisfy the relation [mu]t = [mu]t-B * [delta](b(t)) for all t > 0. B is called an exponent for [mu]. It is proved here that if an operator-stable distribution on Rn has n linearly independent univariate stable marginals, then its exponents are semi-simple operators. In addition necessary and sufficient conditions are given for such a distribution on R2 to have univariate stable marginals. The proofs use a hitherto unpublished result of Sharpes that all full operator-stable distributions are absolutely continuous. His proof is provided here.

Operator stable laws: Series representations and domains of normal attraction

LetX,X1,X2,... be i.i.d. random vectors in ℝd. The limit laws μ that can arise by suitable affine normalizations of the partial sums,Sn=X1+...+Xn, are calledoperator-stable laws. These laws are a natural extension to ℝd of the stable laws onℝ. Thegeneralized domain of attraction of μ[GDOA(μ)] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to μ. If the linear part of the affine transformation is restricted to take the formn−B for some exponent operatorB naturally associated to μ thenX is in thegeneralized domain of normal attraction of μ [GDONA(μ)]. This paper extends the theory of operator-stable laws μ and their domains of attraction and normal attraction.

Limit theorems for variational sums

Limit theorems in the sense of a.s. convergence, convergence in L 1-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if XX(t), 0 < t < T} is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form E1g(X(t,k) X(tn,lk )) is shown to converge with probability one and in L 1-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.

Operator — Stable distributions with independent marginals

SummaryLet Μ be a full operator-stable probability measure over a finite dimensional real inner product space V. Necessary and sufficient conditions are obtained for Μ to have independent univariate marginals with respect to some basis of V. These essentially amount to the statement that the support of the Lévy spectral measure is a subset of the union of one-dimensional subspaces of V determined by the vectors in a basis of the subspace of V spanned by the support of the non-Gaussian component. A representation for an exponent of such a Μ is also given.

Limit theorems for the multivariate binomial distribution

Nonsingular limit distributions are determined for sequences of affine transformations of random vectors whose distributions are multivariate binomial. Each of these limit distributions is that of an affine transformation of a random vector having a multivariate normal distribution or a multivariate Possion distribution or a joint distribution of two independent random vectors, one normal and the other Poisson.