The aim of this manuscript is to determine the relative size of several functions (copulas, quasi–copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi–copulas. Moreover, it is proved that copulas are nowhere dense in the clas...

We continue our study of burn-off chip-firing games on graphs initiated in [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121–132]. Here we introduce randomness by choosing each successive seed uniformly from among all possible nodes. The resulting stochastic process—a Markov chain (Xn)n≥0 with state space the set R of relaxed legal chip configurations C : V → N on a connected graph G = ...

To a unitary matrix U we associate a doubly stochastic matrix M by taking the squared modulus of each element of U . To study the connection between onset of quantum chaos on graphs and ergodicity of the underlying Markov chain, specified by M , we study the limiting distribution of the spectral gap of M when U is taken from the circular unitary ensemble and the dimension N of U is taken to inf...

For the sake of completeness, we first present the Sinkhorn’s theorem and explain how it was used to derive the solution of the regularized optimal transport. Theorem ((Sinkhorn & Knopp, 1967)). If A is an n × n matrix with strictly positive elements, then there exist diagonal matrices D1 and D2 with strictly positive diagonal elements such that D1 ∗ A ∗ D2 is doubly stochastic. The matrices D1...

This paper is concerned with a new type of differential game problems of forward-backward stochastic systems. There are three distinguishing features: Firstly, our game systems are forward-backward doubly stochastic differential equations, which is a class of more general game systems than other forward-backward stochastic game systems without doubly stochastic terms; Secondly, forward equation...