CUTTING A PART FROM MANY MEASURES

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint.We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$ . Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geqslant 2$ , $d\geqslant 2$ , $c\geqslant d$ with $m\geqslant n(c-d)+d$ are integers, and $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^{d}$ , we prove that there exists a partition of $\mathbb{R}^{d}$ into $n$ convex pieces which equiparts the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$ , and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ .

Estimation of integrals with respect to infinite measures using regenerative sequences

Letfbe an integrable function on an infinite measure space (S,, π). We show that if aregenerative sequence{Xn}n≥0with canonical measureπcould be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on ℝdusing a simple symmetric random walk on ℤ.

Estimation of integrals with respect to infinite measures using regenerative sequences

Letfbe an integrable function on an infinite measure space (S,, π). We show that if aregenerative sequence{Xn}n≥0with canonical measureπcould be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on ℝdusing a simple symmetric random walk on ℤ.

On the Rate of Convergence of Empirical Measures in ∞-transportation Distance

We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the ∞-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.

An Extension of Young's Inequality

Young's inequality is extended to the context of absolutely continuous measures. Several applications are included.

FLEMING–VIOT PROCESSES IN AN ENVIRONMENT

We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line ℝ is constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson–Watanabe superprocesses, conditioned to have total mass one, are Fleming–Viot superprocesses.

INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS

A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].