We uncover geometric aspects that underlie the sum of two independent stochastic variables when both are governed by q−Gaussian probability distributions. The pertinent discussion is given in terms of random vectors uniformly distributed on a p−sphere.

Fluctuations of the order parameters of the Gardner model for any α < α c are studied. It is proved that they converge in distribution to a family of jointly Gaussian random variables.

We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result gives the duality between moments of these families and the corresponding real Wigner and Wishart families.

In the paper, we consider the density functions of random variables that can be written as integrals of exponential functions of Gaussian random fields. In particular, we provide closed form asymptotic bounds for the density functions and under smoothness conditions we derive exact tail approximations of the density functions.

We obtain a general form of the multivariate Rayleigh and exponential probability density functions (p.d.f.s) when these are generated by corre lated Gaussian random variables. A general expression for the exponential characteristic function (c.f.) is also derived.

The expectations of random determinants whose entries are real-valued, identically distributed, mean zero, correlated Gaussian random variables are examined using the Kronecker tensor products and some combinatorial arguments. This result is used to derive the expected determinants of X +B and AX +X ′B.

We consider the moment space Mn corresponding to p × p real or complex matrix measures defined on the interval [0, 1]. The asymptotic properties of the first k components of a uniformly distributed vector (S1,n, . . . , Sn,n) ∗ ∼ U(Mn) are studied if n → ∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S1,n, . . . , Sk,n) ∗ converges weakly to...

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Understanding complex relationships between random variables is of fundamental importance in high-dimensional statistics, with numerous applications in biological and social sciences. Undirected graphical models are often used to represent dependencies between random variables, where an edge between to random variables is drawn if they are conditionally dependent given all the other measured va...