62 research outputs found
Free subexponentiality
In this article, we introduce the notion of free subexponentiality, which
extends the notion of subexponentiality in the classical probability setup to
the noncommutative probability spaces under freeness. We show that
distributions with regularly varying tails belong to the class of free
subexponential distributions. This also shows that the partial sums of free
random elements having distributions with regularly varying tails are tail
equivalent to their maximum in the sense of Ben Arous and Voiculescu [Ann.
Probab. 34 (2006) 2037-2059]. The analysis is based on the asymptotic
relationship between the tail of the distribution and the real and the
imaginary parts of the remainder terms in Laurent series expansion of Cauchy
transform, as well as the relationship between the remainder terms in Laurent
series expansions of Cauchy and Voiculescu transforms, when the distribution
has regularly varying tails.Comment: Published in at http://dx.doi.org/10.1214/11-AOP706 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Convergence of joint moments for independent random patterned matrices
It is known that the joint limit distribution of independent Wigner matrices
satisfies a very special asymptotic independence, called freeness. We study the
joint convergence of a few other patterned matrices, providing a framework to
accommodate other joint laws. In particular, the matricial limits of symmetric
circulants and reverse circulants satisfy, respectively, the classical
independence and the half independence. The matricial limits of Toeplitz and
Hankel matrices do not seem to submit to any easy or explicit
independence/dependence notions. Their limits are not independent, free or half
independent.Comment: Published in at http://dx.doi.org/10.1214/10-AOP597 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Extremes of some Gaussian random interfaces
In this article we give a general criterion for some dependent Gaussian
models to belong to maximal domain of attraction of Gumbel, following an
application of the Stein-Chen method studied in Arratia et al(1989). We also
show the convergence of the associated point process. As an application, we
show the conditions are satisfied by some of the well-known supercritical
Gaussian interface models, namely, membrane model, massive and massless
discrete Gaussian free field, fractional Gaussian free field.Comment: To appear in Journal of Statistical Physic
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