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Register a new userA Functional Equation of Tail-balance for Continuous Signals in the Condorcet Jury Theorem
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Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability $p$, then the probability of a correct verdict tends to one as the jury size tends to infinity (Condorcet, 1785). Recently, the first two authors developed a model where jurors sequentially receive signals from an interval according to a distribution, which depends on the state of Nature and on the jurors ability, and vote sequentially. This paper shows that to mimic Condorcets binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio $alpha(t)$ of the probability that a mean-zero random variable satisfies $X >t$ given that $|X|>t$. In particular, we show that under natural symmetry assumptions the tail-balances $alpha(t)$ uniquely determine the distribution.
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We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method of investigations based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.
Let $(Z_n,ngeq 0)$ be a supercritical Galton-Watson process whose offspring distribution $mu$ has mean $lambda>1$ and is such that $int x(log(x))_+ dmu(x)<+infty$. According to the famous Kesten & Stigum theorem, $(Z_n/lambda^n)$ converges almost surely, as $nto+infty$. The limiting random variable has mean~1, and its distribution is characterised as the solution of a fixed point equation. par In this paper, we consider a family of Galton-Watson processes $(Z_n(lambda), ngeq 0)$ defined for~$lambda$ ranging in an interval $Isubset (1, infty)$, and where we interpret $lambda$ as the time (when $n$ is the generation). The number of children of an individual at time~$lambda$ is given by $X(lambda)$, where $(X(lambda))_{lambdain I}$ is a c`adl`ag integer-valued process which is assumed to be almost surely non-decreasing and such that $mathbb E(X(lambda))=lambda >1$ for all $lambdain I$. This allows us to define $Z_n(lambda)$ the number of elements in the $n$th generation at time $lambda$. Set $W_n(lambda)= Z_n(lambda)/lambda^n$ for all $ngeq 0$ and $lambdain I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(lambda), lambdain I)_{ngeq 0}$ converges in probability as~$n$ tends to infinity in the space of c`adl`ag processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.
We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppalainen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_{gamma}$ condition of Sznitman we reduce the moment condition to ${Bbb E}(tau^2(ln tau)^{1+m})<+infty$ for $m>1+1/gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.
Sufficient and necessary conditions are presented for the comparison theorem of path dependent $G$-SDEs. Different from the corresponding study in path independent $G$-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case.
The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.
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