Operator Stable Law Research Articles

Matrix Normalized Sums of Independent Identically Distributed Random Vectors

Let $X_1, X_2,\cdots$ be a sequence of independent identically distributed random vectors and $S_n = X_1 + \cdots + X_n$. Necessary and sufficient conditions are given for there to exist matrices $B_n$ and vectors $\gamma_n$ such that $\{B_n(S_n - \gamma_n)\}$ is stochastically compact, i.e., $\{B_n(S_n - \gamma_n)\}$ is tight and no subsequential limit is degenerate. When this condition holds we are able to obtain precise estimates on the distribution of $S_n$. These results are then specialized to the case where $X_1$ is in the generalized domain of attraction of an operator stable law and a local limit theorem is proved which generalizes the classical local limit theorem where the normalization is done by scalars.

Affine normability of partial sums of I.I.D. random vectors: A characterization

Let X, X1,X2,... be i.i.d. d-dimensional random vectors with partial sums S n . We identify the collection of random vectors X for which there exist non-singular linear operators T n and vectors υn∈ℝ d such that {ℒ(T n (S n −υ n )),n>=1} is tight and has only full weak subsequential limits. The proof is constructive, providing a specific sequence {T n }. The random vector X is said to be in the generalized domain of attraction (GDOA) of a necessarily operator-stable law γ if there exist {T n } and {υ n } such that ℒ(T n (S n −υ n ))→γ. We characterize the GDOA of every operator-stable law, thereby extending previous results of Hahn and Klass; Hudson, Mason, and Veeh; and Jurek. The characterization assumes a particularly nice form in the case of a stable limit. When γ is symmetric stable, all marginals of X must be in the domain of attraction of a stable law. However, if γ is a nonsymmetric stable law then X may be in the GDOA of γ even if no marginal is in the domain of attraction of any law.

The Domain of Normal Attraction of an Operator-Stable Law

The idea of the domain of normal attraction was earlier extended to probabilities on a finite-dimensional inner-product space. We obtain a necessary and sufficient condition that a probability be in the domain of normal attraction of a given probability in terms of their covariance operators and of a limit involving the Levy measure. This condition appears to be the natural generalization of the corresponding univariate condition. We also show that the domains of normal attraction of two probabilities are either the same or disjoint, with a condition that is necessary and sufficient for them to be the same.

Operator-Stable Laws: Multiple Exponents and Elliptical Symmetry

We characterize the class of linear operators on a finite dimensional inner product space which are the exponents of a full operator-stable law. This answers a question of Paulauskas [6] concerning those operator-stable laws whose characteristic functions are the exponential of quadratic forms. The symmetry group of such laws must be conjugate to the group of all orthogonal transformations on the space.