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Register a new userFixed points with finite mean of the smoothing transform in random environments
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At each time $ninmathbb{N}$, let $bar{Y}^{(n)}=(y_{1}^{(n)},y_{2}^{(n)},cdots)$ be a random sequence of non-negative numbers that are ultimately zero in a random environment $xi=(xi_{n})_{ninmathbb{N}}$ in time, which satisfies for each $ninmathbb{N}$ and a.e. $xi,~E_{xi}[sum_{iinmathbb{N}_{+}}y_{i}^{(n)}(xi)]=1.$ The existence and uniqueness of the non-negative fixed points of the associated smoothing transform in random environments is considered. These fixed points are solutions of the distributional equation for $a.e.~xi,~Z(xi)overset{d}{=}sum_{iinmathbb{N}_{+}}y_{i}^{(0)}(xi)Z_{i}(Txi),$ where when given the environment $xi$, $Z_{i}(Txi)~(iinmathbb{N}_{+})$ are $i.i.d.$ non-negative random variables, and distributed the same as $Z(xi)$. As an application, the martingale convergence of the branching random walk in random environments is given as well. The classical results by Biggins (1977) has been extended to the random environment situation.
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We consider vector fixed point (FP) equations in large dimensional spaces involving random variables, and study their realization-wise solutions. We have an underlying directed random graph, that defines the connections between various components of the FP equations. Existence of an edge between nodes i, j implies the i th FP equation depends on the j th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random neighbor components. We obtain finite dimensional limit FP equations (in a much smaller dimensional space), whose solutions approximate the solution of the random FP equations for almost all realizations, in the asymptotic limit (number of components increase). Our techniques are different from the traditional mean-field methods, which deal with stochastic FP equations in the space of distributions to describe the stationary distributions of the systems. In contrast our focus is on realization-wise FP solutions. We apply the results to study systemic risk in a large financial heterogeneous network with many small institutions and one big institution, and demonstrate some interesting phenomenon.
Let $mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.
For an element $g$ of a group $G$, an Engel sink is a subset $mathscr{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathscr{E}(g)$. Let $q$ be a prime, let $m$ be a positive integer and $A$ an elementary abelian group of order $q^2$ acting coprimely on a finite group $G$. We show that if for each nontrivial element $a$ in $ A$ and every element $gin C_{G}(a)$ the cardinality of the smallest Engel sink $mathscr{E}(g)$ is at most $m$, then the order of $gamma_infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $ain Asetminus {1}$ and every element $gin C_{G}(a)$, the smallest Engel sink $mathscr{E}(g)$ generates a subgroup of rank at most $m$, then the rank of $gamma_infty(G)$ is bounded in terms of $m$ and $q$ only.
We give necessary and sufficient conditions for a function in a naturally appearing functional space to be a fixed point of the Ruelle-Thurston operator associated to a rational function, see Lemma 2.1. The proof uses essentially a recent [13]. As an immediate consequence, we revisit Theorem 1 and Lemma 5.2 of [11], see Theorem 1 and Lemma 2.2 below.
We consider the sums $S_n=xi_1+cdots+xi_n$ of independent identically distributed random variables. We do not assume that the $xi$s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${bf P}{M>x}$ as $xtoinfty$, provided that $M=sup{S_n, nge1}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that subexponentiality of distribution $F$ does not imply subexponentiality of its integrated tail distribution $F^I$.
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