Consider an election between two candidates in which the voters' choices are random and independent and the probability of a voter choosing the first candidate is $p>1/2$. Condorcet's Jury Theorem which he derived from the weak law of large numbers asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. The notion of influence of a voter or its voting power is relevant for extensions of the weak law of large numbers for voting rules which are more general than simple majority. In this paper we point out two different ways to extend the classical notions of voting power and influences to arbitrary probability distributions. The extension relevant to us is the ``effect'' of a voter, which is a weighted version of the correlation between the voter's vote and the election's outcomes. We prove an extension of the weak law of large numbers to weighted majority games when all individual effects are small and show that this result does not apply to any voting rule which is not based on weighted majority.
Benjamini，Kalai and Schramm（2001）证明了$n$独立无偏比特的加权多数函数在噪声下是一致稳定的：当每个比特以概率$ε$翻转时，加权多数改变的概率$p_ε$至多为$Cε^{1/4}$。他们问什么是可以取代1/4的最佳指数。我们证明答案是1/2。$p_ε$的上界与$ε\to 0$和$nε\to \infty$的已知下界相差$\sqrt{π/2}+o(1)$。
Crowdsourcing has become an effective and popular tool for human-powered computation to label large datasets. Since the workers can be unreliable, it is common in crowdsourcing to assign multiple workers to one task, and to aggregate the labels in order to obtain results of high quality. In this paper, we provide finite-sample exponential bounds on the error rate (in probability and in expectation) of general aggregation rules under the Dawid-Skene crowdsourcing model. The bounds are derived for multi-class labeling, and can be used to analyze many aggregation methods, including majority votin…
A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted one-sided ergodic Hilbert transforms on vector-valued $L_p$-spaces. Furthermore, the obtained results are applied to the a.e. convergence of random modulated weighted ergodic series, which is also new in the classical setting.
Given a graph $G$ and some initial labelling $σ: V(G) \to \{Red, Blue\}$ of its vertices, the \textit{majority dynamics model} is the deterministic process where at each stage, every vertex simultaneously replaces its label with the majority label among its neighbors (remaining unchanged in the case of a tie). We prove---for a wide range of parameters---that if an initial assignment is fixed and we independently sample an Erdős--Rényi random graph, $G_{n,p}$, then after one step of majority dynamics, the number of vertices of each label follows a central limit law. As a corollary, we provide a…
We consider weighted sums of independent random variables regulated by an increment sequence. We provide operative conditions that ensure strong law of large numbers for such sums to hold in both the centered and non-centered case. The existing criteria for the strong law are either implicit or assume some sufficient decay for the sequence of coefficients. In our set up we allow for arbitrary sequence of coefficients, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centered case, convergence can be translated in…
We consider complex Mandelbrot multiplicative cascades on a random weigh\-ted tree. Under suitable assumptions, this yields a dynamics $\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e., sum up to~1 almost surely, we find a domain for the initial law $μ$ such that a non-standard (functional) central limit theorem is valid for the orbit $(\T^nμ)_{n\ge 0}$ (this completes in a non trivial way our previous result in th…
We consider a weighted lattice $Z^d$ with conductance $μ_e=|e|^{-α}$. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when $d=2$ and $α\in (-1,0)$, two independent random walks on such weighted lattice will collide infinite many times while they are transient.
For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $$ \mathfrak C_N(T)= \mathbb E\bigg[ \prod_{(i,j) \in T} \Big(\mathbf 1_{\big(\{i,j\} \in G_N \big)} - \mathbb P(\{i,j\} \in G_N) \Big)\bigg], \ T \subset [N]^2 \textrm{finite}$$ furnishes informations on the asymptotic spectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N = N\times \mathbb P(\{i,j\} \in G_N) $ and assume $d_N, N-d_N\underset{N \righta…
本文研究了加权随机和的渐近性态，当和过程按稳定律收敛于布朗运动，且加权过程具有比布朗运动更规则的连续轨线时.我们证明了当观测值之间的距离趋于零时，这些和收敛于关于布朗运动的权过程的积分。结果是通过分数阶微积分的帮助下得到的，显示了这种技术的力量。这项研究，虽然有趣的本身，是由一个错误的证明定理4中发现的Corcuera，J.M. Nualart，D.，Woerner，J. H. C.（2006年）。一些积分分数过程的幂变分，伯努利12（4）713-735。
In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the Marcinkiewich-Zygmund type and Kolmogorov type strong laws of large numbers are derived. In addition, as applications of our results, Stranssen type invariance principles of negatively associated random variables and vertically independent random variables are proposed respectively.
The aim of this note is to prove a law of large numbers for local patterns in discrete point processes. We investigate two different situations: a class of point processes on the one dimensional lattice including certain Schur measures, and a model of random plane partitions, introduced by Okounkov and Reshetikhin. The results state in both cases that the linear statistic of a function, weighted by the appearance of a fixed pattern in the random configuration and conveniently normalized, converges to the deterministic integral of that function weighted by the expectation with respect to the li…
