We show that for many models of random trees, the independence number divided by the size converges almost surely to a constant as the size grows to infinity; the trees that we consider include random recursive trees, binary and $m$-ary search trees, preferential attachment trees, and others. The limiting constant is computed, analytically or numerically, for several examples. The method is based on Crump-Mode-Jagers branching processes.
We are interested in the independence number of large random simply generated trees and related parameters, such as their matching number or the kernel dimension of their adjacency matrix. We express these quantities using a canonical tricolouration, which is a way to colour the vertices of a tree with three colours. As an application we obtain limit theorems in $L^p$ for the renormalised independence number in large simply generated trees (including large size-conditioned Bienaymé-Galton-Watson trees).
We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.
The paper develops a rather unexpected parallel to the multivariate Matsumoto--Yor (MY) property on trees considered in \cite{MW04}. The parallel concerns a multivariate version of the Kummer distribution, which is generated by a tree. Given a tree of size $p$, we direct it by choosing a vertex, say $r$, as a root. With such a directed tree we associate a map $Φ_r$. For a random vector ${\bf S}$ having a $p$-variate tree-Kummer distribution and any root $r$, we prove that $Φ_r({\bf S})$ has independent components. Moreover, we show that if ${\bf S}$ is a random vector in $(0,\infty)^p$ and for…
In this paper we construct the new coefficient which allows to measure quantitatively the independence of the two discrete random variables. The new inequalities for the matrices with non-negative elements are found
There is given a characterization of the geometric distribution by the independence of linear forms with random coefficients. The result is a discrete analog of the corresponding theorem on exponential distribution. The property of linear statistics independence is also a characterization of Poisson law. Keywords: geometric distribution; exponential distribution; Poisson distribution; linear forms; random coefficients
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this fact is now superseded by a general result.
We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of the $N$-regular tree, we define the $\mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the ope…
We give conditions under which a scalar random variable T can be coupled to a random scaling factor $ξ$ such that T and $ξ$T are rendered stochastically independent. A similar result is obtained for random measures. One consequence is a generalization of a result by Pitman and Yor on the Poisson-Dirichlet distribution to its negative parameter range. Another application are diffusion excursions straddling an exponential random time.
It is shown that a Wishart matrix of standard complex normal random variables is asymptotically freely independent of an independent random matrix, under minimal conditions, in two different sense of asymptotic free independence.
Let $\{ξ(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables with conditions of type $D(u_n)$ and $D'(u_n)$. Let $\{S_n, n \in \mathbb{N} \}$ be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first $n$ terms of the sequence $\{ξ(S_n), n \in \mathbb{N} \}$ as $n$ goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence $\{ξ(k), k \in \mathbb{Z} \}$ is i.i.d.
For a sequence in discrete time having stationary independent values (respectively, random walk) $X$, those random times $R$ of $X$ are characterized set-theoretically, for which the strict post-$R$ sequence (respectively, the process of the increments of $X$ after $R$) is independent of the history up to $R$. For a Lévy process $X$ and a random time $R$ of $X$, reasonably useful sufficient conditions and a partial necessary condition on $R$ are given, for the process of the increments of $X$ after $R$ to be independent of the history up to $R$.
A collection of $n$ random events is said to be $(n - 1)$-wise independent if any $n - 1$ events among them are mutually independent. We characterise all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent. We provide sharp upper and lower bounds on the probability that at least $k$ out of $n$ events with given marginal probabilities occur over these probability measures. The bounds are shown to be computable in polynomial time.
A rearrangement of $n$ independent uniform $[0,1]$ random variables is a sequence of $n$ random variables $Y_1,...,Y_n$ whose vector of order statistics has the same distribution as that for the $n$ uniforms. We consider rearrangements satisfying the strong rank independence condition, that the rank of $Y_k$ among $Y_1,...,Y_k$ is independent of the values of $Y_1,...,Y_{k-1}$, for $k=1,...,n$. Nontrivial examples of such rearrangements are the travellers' processes defined by Gnedin and Krengel. We show that these are the only examples when $n=2$, and when certain restrictive assumptions hold…
We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and…
Conditioned limit laws constitute an important and well developed framework of extreme value theory that describe a broad range of extremal dependence forms including asymptotic independence. We explore the assumption of conditional independence of $X_1$ and $X_2$ given $X_0$ and study its implication in the limiting distribution of $(X_1,X_2)$ conditionally on $X_0$ being large. We show that under random norming, conditional independence is always preserved in the conditioned limit law but might fail to do so when the normalisation does not include the precise value of the random variable in …
Prakasa Rao (Studia Sci. Math. Hungar. 11 (1976) 277-282) studied a characterization of probability distributions for linear functions of independent generalized random fields. These results are extended to Q-independent generalized random fields. It is known that independence implies Q-independence but the converse is not true.
We count the number and patterns of pairs and tuples of independent events in a simple random experiment: first a fair coin is flipped and then a fair die is tossed. The first number, equal to 888,888, suggest that there are some open questions about the structure of independence even in a finite sample space. We discuss briefly these questions and possible approaches to answer them.
Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures
We construct several new spaces of quantum sequences and their quantum families of maps in sense of Sołtan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by inte…
We show that there can be no finite list of conditional independence relations which can be used to deduce all conditional independence implications among Gaussian random variables. To do this, we construct, for each $n> 3$ a family of $n$ conditional independence statements on $n$ random variables which together imply that $X_1 \ind X_2$, and such that no subset have this same implication. The proof relies on binomial primary decomposition.
The Freund family of distributions becomes a Riemannian 4-manifold with Fisher information as metric; we derive the induced $α$-geometry, i.e., the $α$-curvature, $α$-Ricci curvature with its eigenvales and eigenvectors, the $α$-scalar curvature etc. We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere. We consider special cases as submanifolds and discuss their geometrical structures; one submanifold yields examples of neighbourhoods of the independent case for bivariate distributions having identical exponential margina…
It is well known that when a pair of random variables is statistically independent, it has no-correlation (zero covariance, $E[XY] - E[X]E[Y] = 0$), and that the converse is not true. However, if both of these random variables take only two values, no-correlation entails statistical independence. We provide here a general proof. We subsequently examine whether this equivalence property carries over to quantum mechanical systems. A counter-example is explicitly constructed to show that it does not. This observation provides yet another simple theorem separating classical and quantum theories.
We study an algorithm for approximating the multivariate independence polynomial $Z(\mathbf{z})$, with negative and complex arguments, an object that has strong connections to combinatorics and to statistical physics. In particular, the independence polynomial with negative arguments, $Z(-\mathbf{p})$, determines the Shearer region, the maximal region of probabilities to which the Lovasz Local Lemma (LLL) can be extended (Shearer 1985). In statistical physics, complex zeros of the independence polynomial relate to existence of phase transitions. Our main result is a deterministic algorithm to …
Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. We first establish this construction for random matrices. We define the traffic distribution of random matrices, which is richer than the *-distribution of free probability. The knowledge of the individual traffic distribut…
Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on $N\times N$ random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix $U_N$. In this article we elaborate on the previous point by proposing a random matrix model, which we name the Vortex model, where $U_N$ has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector $v_N$. In the limit $N \to +\infty$, we show that $N\times N$ matrices randomly rotated by the matrix $U_N$ are asymptotically conditionally f…
Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $λ$ tending to infinity. Finding the independence ratio of random $d$-regular graphs for some fixed degree $d$ has received much attention both in random graph theory and in statistical physics. For $d \geq 20$ the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method s…
Kagan and Shalaevski 1967 have shown that if the random variables $X_1,\dots,X_n$ are independent and identically distributed and the distribution of $\sum_{i=1}^n(X_i+a_i)^2$ $a_i\in \mathbb{R}$ depends only on $\sum_{i=1}^na_i^2$ , then each $X_i$ follows the normal distribution $N(0, σ)$. Cook 1971 generalized this result replacing independence of all $X_i$ by the independence of $(X_1,\dots, X_m) \textrm{ and } (X_{m+1},\dots,X_n )$ and removing the requirement that $X_i$ have the same distribution. In this paper, we will give other characterizations of the normal distribution which are fo…
A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $λ_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least \[q=\frac{1}{2}-\frac{3}{4π}\arccos\biggl(\frac{1-λ_{\min}}{4}\biggr).\] The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is…
We study the independence structure of finitely exchangeable distributions over random vectors and random networks. In particular, we provide necessary and sufficient conditions for an exchangeable vector so that its elements are completely independent or completely dependent. We also provide a sufficient condition for an exchangeable vector so that its elements are marginally independent. We then generalize these results and conditions for exchangeable random networks. In this case, it is demonstrated that the situation is more complex. We show that the independence structure of exchangeable …
The standard method to check for the independence of two real-valued random variables -- demonstrating that the bivariate joint distribution factors into the product of its marginals -- is both necessary and sufficient. Here we present a simple necessary condition based on the support sets of the random variables, which -- if not satisfied -- avoids the need to extract the marginals from the joint in demonstrating dependence. We review, in an accessible manner, the measure-theoretic, topological, and probabilistic details necessary to establish the background for the old and new ideas presente…
In this note we prove a large deviation bound on the sum of random variables with the following dependency structure: there is a dependency graph $G$ with a bounded chromatic number, in which each vertex represents a random variable. Variables that are represented by neighboring vertices may be arbitrarily dependent, but collections of variables that form an independent set in $G$ are $t$-wise independent.
We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$ time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. For a determinantal point process on subsets of size $k$ of a ground set of $n$ elements, we show how to approximately sample in $\widetilde{O}(k^ω)$ time after an initial $\widetilde{O}(nk^{ω-1})$ time prep…
We consider the first-order theory of random variables with the probabilistic independence relation, which concerns statements consisting of random variables, the probabilistic independence symbol, logical operators, and existential and universal quantifiers. Although probabilistic independence is the only non-logical relation included, this theory is surprisingly expressive, and is able to interpret the true first-order arithmetic over natural numbers (and hence is undecidable). We also construct a single-letter characterization of the capacity region for a general class of multiuser coding s…
