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Register a new userLift zonoid and barycentric representation on a Banach space with a cylinder measure
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We show that the lift zonoid concept for a probability measure on R^d, introduced in (Koshevoy and Mosler, 1997), leads naturally to a one-to one representation of any interior point of the convex hull of the support of a continuous measure as the barycenter w.r.t. to this measure of either of a half-space, or the whole space. We prove the infinite-dimensional generalization of this representation, which is based on the extension of the lift-zonoid concept for a cylindrical probability measure.
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Let $A$ be a real commutative Banach algebra with unity. Let $a_0in Asetminus{0}$. Let $mathbb Z a_0:={na_0}_{nin mathbb Z}$. Then, $mathbb Z a_0$ is a discrete subgroup of $A$. For any $nin mathbb Z$, the Frechet derivative of the mapping $$x , in , A mapsto x+na_0 , in , A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x , in , A mapsto x+mathbb Z a_0 , in , A/(mathbb Z a_0)$$ is an $A$-map. We call $A/(mathbb Z a_0)$ the $1$-dimensional $A$-cylinder by $a_0$. Let $T$ be a compact Hausdorff space. Suppose that there exist $t_1in T$ and $t_2in T$ such that $t_1 ot=t_2$ holds. Then, the set $C(T;mathbb R)$ of all real-valued continuous functions on $T$ is a real commutative Banach algebra with unity and $mathbb R , subsetneq , C(T;mathbb R)$ holds. In this paper, we show that there exists $a_0 , in , C(T;mathbb R)setminus mathbb R$ such that for any $k, in , mathbb N$, the $1$-dimensional $C(T;mathbb R)$-cylinder $(C(T;mathbb R))/(mathbb Z a_0)$ by $a_0$ cannot be embedded in the finite direct product space $(C(T;mathbb R))^k$ as a $C(T;mathbb R)$-submanifold.
We give a generalization to a continuous setting of the classic Markov chain tree Theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices and an absolutely continuous part on the edges. We show that the corresponding density at $x$ can be represented by a normalized superposition of the weights associated to metric arborescences oriented toward the point $x$. The weight of each oriented metric arborescence is obtained by the exponential of integrals of the form $intfrac{b}{sigma^2}$ along the oriented edges time a weight for each node determined by the local orientation of the arborescence around the node time the inverse of the diffusion coefficient at $x$. The metric arborescences are obtained cutting the original metric graph along some edges.
In this paper, we present a mixed-type integral-sum representation of the cylinder functions $mathscr{C}_mu(z)$, which holds for unrestricted complex values of the order $mu$ and for any complex value of the variable $z$. Particular cases of these representations and some applications, which include the discussion of limiting forms and representations of related functions, are also discussed.
We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $beta$, and then give convergence theorems for martingales of $beta$-conditional expectations. We give the Birkhoff ergodic theorem for $beta$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $beta$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on $M$. Finally, the large derivation property of $beta$-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.
The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element $zeta$ of a certain class of c`{a}dl`{a}g paths that take values in the space of signed measures on the half-line to a c`{a}dl`{a}g path that takes values in the space of non-negative measures on $[0,infty)$ in such a way that for each $x > 0$, the path $t mapsto zeta_t[0,x]$ is transformed via a Skorokhod map on the half-line, and the regulating functions for different $x > 0$ are coupled. We establish regularity properties of this map and show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well known models are the earliest-deadline-first, the shortest-job-first and the shortest-remaining-processing-time scheduling policies. For these applications, we show how the map provides a unified framework within which to form fluid model equations, prove uniqueness of solutions to these equations and establish convergence of scaled state processes to the fluid model. In particular, for these models, we obtain new convergence results in time-inhomogeneous settings, which appear to fall outside the purview of existing approaches.
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