### On Choquet's theory

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Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda =\rho \left(A\right),{\lambda}_{2},...,{\lambda}_{n}$. Fiedler and others have shown that $det(\lambda I-A)\le {\lambda}^{n}-{\rho}^{n}$, for all $\lambda >\rho $, with equality for any such $\lambda $ if and only if $A$ is the simple cycle matrix. Let ${a}_{i}$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i=1,...,n-1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $det(\lambda I-A)+{\sum}_{i=1}^{n-1}{\rho}^{n-2i}\left|{a}_{i}\right|{(\lambda -\rho )}^{i}\le {\lambda}^{n}-{\rho}^{n}$, for all $\lambda \ge \rho $. We use this inequality to derive the inequality that: ${\prod}_{2}^{n}(\rho -{\lambda}_{i})\le {\rho}^{n-2}{\sum}_{i=2}^{n}(\rho -{\lambda}_{i})$. In the spirit of a celebrated conjecture due to Boyle-Handelman,...

In a recent paper the authors proposed a lower bound on $1-{\lambda}_{i}$, where ${\lambda}_{i}$, ${\lambda}_{i}\ne 1$, is an eigenvalue of a transition matrix $T$ of an ergodic Markov chain. The bound, which involved the group inverse of $I-T$, was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when...

Let $A$ be an $n\times n$ symmetric, irreducible, and nonnegative matrix whose eigenvalues are ${\lambda}_{1}>{\lambda}_{2}\ge ...\ge {\lambda}_{n}$. In this paper we derive several lower and upper bounds, in particular on ${\lambda}_{2}$ and ${\lambda}_{n}$, but also, indirectly, on $\mu ={max}_{2\le i\le n}\left|{\lambda}_{i}\right|$. The bounds are in terms of the diagonal entries of the group generalized inverse, ${Q}^{\#}$, of the singular and irreducible M-matrix $Q={\lambda}_{1}I-A$. Our starting point is a spectral resolution for ${Q}^{\#}$. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...

We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

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