Related papers: On the Lukacs property for free random variables
Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$. We also prove a similar result for the $k$-free values of…
Applying the concept of matricial freeness which generalizes freeness in free probability, we have recently studied asymptotic joint distributions of symmetric blocks of Gaussian random matrices (Gaussian Symmetric Block Ensemble). This…
For a general free L\'evy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the L\'evy-It\^o representation of the original process. For a general free compound…
The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $\alpha$ and…
This work investigates the intersection property of conditional independence. It states that for random variables $A,B,C$ and $X$ we have that $X$ independent of $A$ given $B,C$ and $X$ independent of $B$ given $A,C$ implies $X$ independent…
Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including…
We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have…
We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation whose distribution is invariant under conjugation. These results were first established to be applied in…
Let $X$ and $Y$ be two real-valued random variables. Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be independent identically distributed copies of $(X,Y)$. Suppose there are two players A and B. Player A has access to $X_{1},X_{2},\ldots$ and…
We study free infinite divisibility (FID) for a class which is called generalized power distributions with free Poisson term by using a complex analytic technique and a calculation for the free cumulants and Hankel determinants. In…
We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same light-tailed subexponential distribution. The examples of a Poisson and geometric number of summands serve as an…
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…
We investigate analytical properties of free stable distributions and discover many connections with their classical counterparts. Our main result is an explicit formula for the Mellin transform, which leads to explicit series…
We investigate commutators of free variables of the form \( i[x, s] \), where \( s \) is a semicircular element. We show that although \( s \) and \( i[x, s] \) are not free, their sum nevertheless satisfies the free additive convolution…
Although the specification of bivariate probability models using a collection of assumed conditional distributions is not a novel concept, it has received considerable attention in the last decade. In this study, a bivariate…
The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional…
If $X$ and $Y$ are independent random variables with distributions $\mu$ and $\nu$ then $U=\psi(X,Y)$ and $V=\phi(X,Y)$ are also independent for some $\psi$ and $\phi$. Properties of this type are known for many important probability…
The main result of this paper states that for independent random variables $X, Y$ taking values in a compact metrisable abelian group, $X + Y$ has the same distribution as $X$, if and only if there exists a compact subgroup $A$ such that…
We derive a formula for the moments and the free cumulants of the multiplication of $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge…
A basic result is that the sample variance for i.i.d. observations is an unbiased estimator of the variance of the underlying distribution (see for instance Casella and Berger (2002)). But what happens if the observations are neither…