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We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees.
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Let $M^sharp_n(mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(mathbb{R})$ class-many times. We characterize the sets of reals which are $Sigma_1$-definable from $mathbb{R}$ over $M_n(mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Pi^1_{n+1}$; (2) for odd $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Sigma^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $mathbb{R}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists for all $ninmathbb{N}$, and that determinacy of all projective games of length $omega^2$ with moves in $mathbb{N}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists and satisfies $mathsf{AD}$ for all $ninmathbb{N}$.
N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $mathbb R$ so that no infinite sumset $X+X={x+y:x,yin X}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:mathbb Rto r$ with $r$ finite there is an infinite $Xsubseteq mathbb R$ so that $c$ is constant on $X+X$.
We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming-Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson-Watanabe model, and the data are collected according to a Poisson point process with intensity given by the process state at discrete times. Both driving processes are diffusions taking values in the space of discrete measures whose support varies with time, and are stationary and reversible with respect to Dirichlet and gamma priors respectively. A common methodology is developed to obtain in closed form the time-marginal posteriors given past and present data. These are shown to belong to classes of finite mixtures of Dirichlet processes and gamma random measures for the two models respectively, yielding conjugacy of these classes to the type of data we consider. We provide explicit results on the parameters of the mixture components and on the mixing weights, which are time-varying and drive the mixtures towards the respective priors in absence of further data. Explicit algorithms are provided to recursively compute the parameters of the mixtures. Our results are based on the projective properties of the signals and on certain duality properties of their projections.
Nonparametric latent structure models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of $d ge 2$ random measures models the distribution of a group of exchangeable observations, while their dependence structure regulates the borrowing of information across different groups. Recent work has quantified the dependence between random measures in terms of Wasserstein distance from the maximally dependent scenario when $d=2$. By solving an intriguing max-min problem we are now able to define a Wasserstein index of dependence $I_mathcal{W}$ with the following properties: (i) it simultaneously quantifies the dependence of $d ge 2$ random measures; (ii) it takes values in [0,1]; (iii) it attains the extreme values ${0,1}$ under independence and complete dependence, respectively; (iv) since it is defined in terms of the underlying Levy measures, it is possible to evaluate it numerically in many Bayesian nonparametric models for partially exchangeable data.
We introduce a probability distribution on $mathcal{P}([0,1]^d)$, the space of all Borel probability measures on $[0,1]^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero lower quasi-Assouad dimension (hence the upper and lower Assouad dimensions are almost surely infinite or zero). We also indicate how the results extend to other Assouad-like dimensions.
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