Authors:J. Domínguez-Molina and Alfonso Rocha-Arteaga
The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible
distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under
this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models
considered in Benaych-Georges  who introduced the models through their laws. Second, random matrix models for free infinitely
divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding
classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations
of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix
models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free
selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ∫0∞e−1dΨtd, d ≥ 1, where Ψtd is a d × d matrix-valued Lévy process satisfying an Ilog condition.
Hungarian has a number of apparently synonymous time adverbs that can measure the duration of time intervals. The paper explores these adverbs in some detail, and argues that contrary to appearances, none ofthem are freely interchangeable. The starting point is a discussion of the property of homogeneity that time adverbs are sensitive to. The paper argues for a specific treatment of homogeneity and a preliminary adverb definition based on that treatment. It is proposed that some, but not all, Hungarian time adverbs share the default definition. The diverging adverbs may (a) contain a covert frequency predicate or (b) not measure the duration of the time interval directly, but by determining an endpoint of the interval. Hungarian time adverbs also differ in the range of time intervals they can measure; some, but not all adverbs can measure all available time intervals including the event, iterative, habitual and reference time. This variability in time adverb modification is arbitrary and needs to be explicitly determined for each adverb. Apart from discerning the interpretation of Hungarian time adverbs, the conclusions have a more general impact. On the one hand, apparently homogeneous adverbs can have disparate definitions. On the other, it is necessary to permit explicit, arbitrary constraints on adverbial modification. It is also argued that time adverbs can impose non-local restrictions on the eventuality modified, strengthening the need for a powerful theory of adverbial modification.