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Nevena Maric

Publications and source records attributed to Nevena Maric.

9 recordsLinked to original sources
arXiv2018

Cut polytope has vertices on a line

The cut polytope ${\rm CUT}(n)$ is the convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation with admissible correlations of symmetric Bernoulli random variables. That probabilistic interpretation is a starting point of this work in conjunction with a natural binary encoding of the CUT($n$). We show that for any $n$, with appropriate scaling, all vertices of the polytope ${\mathbf 1}$-CUT($n$) encoded as integers are approximately on the line $y= x-1/2$.

cs.DM↗
arXiv2017

Bernoulli Correlations and Cut Polytopes

Given $n$ symmetric Bernoulli variables, what can be said about their correlation matrix viewed as a vector? We show that the set of those vectors $R(\mathcal{B}_n)$ is a polytope and identify its vertices. Those extreme points correspond to correlation vectors associated to the discrete uniform distributions on diagonals of the cube $[0,1]^n$. We also show that the polytope is affinely isomorphic to a well-known cut polytope ${\rm CUT}(n)$ which is defined as a convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. The isomorphism is obtained explicitly as $R(\mathcal{B}_n)= {\mathbf{1}}-2~{\rm CUT}(n)$. As a corollary of this work, it is straightforward using linear programming to determine if a particular correlation matrix is realizable or not. Furthermore, a sampling method for multivariate symmetric Bernoullis with given correlation is obtained. In some cases the method can also be used for general, not exclusively Bernoulli, marginals.

math.PR↗
arXiv2016

Multivariate distributions with fixed marginals and correlations

Consider the problem of drawing random variates $(X_1,\ldots,X_n)$ from a distribution where the marginal of each $X_i$ is specified, as well as the correlation between every pair $X_i$ and $X_j$. For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between $X_i$ and $X_j$. Any achievable correlation between $X_i$ and $X_j$ is a convex combinations of these bounds. The value $λ(X_i,X_j) \in [0,1]$ of this convex combination is called here the convexity parameter of $(X_i,X_j),$ with $λ(X_i,X_j) = 1$ corresponding to the upper bound and maximal correlation. For given marginal distributions functions $F_1,\ldots,F_n$ of $(X_1,\ldots,X_n)$ we show that $λ(X_i,X_j) = λ_{ij}$ if and only if there exist symmetric Bernoulli random variables $(B_1,\ldots,B_n)$ (that is $\{0,1\}$ random variables with mean 1/2) such that $λ(B_i,B_j) = λ_{ij}$. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three and four dimensions.

math.PR↗
arXiv2014

Minimum correlation for any bivariate Geometric distribution

Consider a bivariate Geometric random variable where the first component has parameter $p_1$ and the second parameter $p_2$. It is not possible to make the correlation between the marginals equal to -1. Here the properties of this minimum correlation are studied both numerically and analytically. It is shown that the minimum correlation can be computed exactly in time $O(p_1^{-1} \ln(p_2^{-1}) + p_2^{-1} \ln(p_1^{-1}))$. The minimum correlation is shown to be nonmonotonic in $p_1$ and $p_2$, moreover, the partial derivatives are not continuous. For $p_1 = p_2$, these discontinuities are characterized completely and shown to lie near (1- roots of 1/2). In addition, we construct analytical bounds on the minimum correlation.

math.PR↗
arXiv2014

Fleming-Viot particle system driven by a random walk on $\mathbb{N}$

Random walk on $\mathbb{N}$ with negative drift and absorption at 0, when conditioned on survival, has uncountably many invariant measures (quasi-stationary distributions, qsd) $ν_c$. We study a Fleming-Viot(FV) particle system driven by this process and show that mean normalized densities of the FV unique stationary measure converge to the minimal qsd, $ν_0$, as $N \to \infty$. Furthermore, every other qsd of the random walk ($ν_c$, $c>0$) corresponds to a metastable state of the FV particle system.

cond-mat.stat-mech↗
arXiv2013

On minimum correlation in construction of multivariate distributions

In this paper we present a method for exact generation of multivariate samples with pre-specified marginal distributions and a given correlation matrix, based on a mixture of Fréchet-Hoeffding bounds and marginal products. The bivariate algorithm can accommodate any among the theoretically possible correlation coefficients, and explicitly provides a connection between simulation and the minimum correlation attainable for different distribution families. We calculate the minimum correlations in several common distributional examples, including in some that have not been looked at before. As an illustration, we provide the details and results of implementing the algorithm for generating three-dimensional negatively and positively correlated Beta random variables, making it the only non-copula algorithm for correlated Beta simulation in dimensions greater than two. This work has potential for impact in a variety of fields where simulation of multivariate stochastic components is desired.

math.PR↗
arXiv2009

Contact process in a wedge

We prove that the supercritical one-dimensional contact process survives in certain wedge-like space-time regions, and that when it survives it couples with the unrestricted contact process started from its upper invariant measure. As an application we show that a type of weak coexistence is possible in the nearest-neighbor ``grass-bushes-trees'' successional model introduced in Durrett and Swindle (1991).

math.PR↗
arXiv2006

Quasi stationary distributions and Fleming-Viot processes in countable spaces

We consider an irreducible pure jump Markov process with rates Q=(q(x,y)) on Λ\cup\{0\} with Λcountable and 0 an absorbing state. A quasi-stationary distribution (qsd) is a probability measure νon Λthat satisfies: starting with ν, the conditional distribution at time t, given that at time t the process has not been absorbed, is still ν. That is, ν(x) = νP_t(x)/(\sum_{y\inΛ}νP_t(y)), with P_t the transition probabilities for the process with rates Q. A Fleming-Viot (fv) process is a system of N particles moving in Λ. Each particle moves independently with rates Q until it hits the absorbing state 0; but then instantaneously chooses one of the N-1 particles remaining in Λand jumps to its position. Between absorptions each particle moves with rates Q independently. Under the condition α:=\sum_x\inf Q(\cdot,x) > \sup Q(\cdot,0):=C we prove existence of qsd for Q; uniqueness has been proven by Jacka and Roberts. When α>0 the {\fv} process is ergodic for each N. Under α>C the mean normalized densities of the fv unique stationary measure converge to the qsd of Q, as N \to \infty; in this limit the variances vanish.

math.PR↗