[Paper Review] On convex functions on the duals of $\Delta_2$-Orlicz spaces

Freddy Delbaen, Keita Owari|arXiv (Cornell University)|Nov 1, 2016

Risk and Portfolio Optimization3 citations

TL;DR

This paper establishes that convex functions on the dual of a $Δ_2$-Orlicz space are lower semicontinuous (resp. continuous) for the Mackey topology if and only if they are lower semicontinuous (resp. continuous) on order intervals with respect to convergence in probability. A key Komlós-type result ensures almost sure convergence of convex combinations of norm-bounded sequences in $L^{\Phi^*}$, with uniform integrability in the Orlicz norm.

ABSTRACT

In the dual $L^{\\Phi^*}$ of a $\\Delta_2$-Orlicz space $L^\\Phi$, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology $\ au(L^{\\Phi^*},L^\\Phi)$ if and only if on each order interval $[-\\zeta,\\zeta]=\\{\\xi: -\\zeta\\leq \\xi\\leq\\zeta\\}$ ($\\zeta\\in L^{\\Phi^*}$), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Koml\\'os type result: every norm bounded sequence $(\\xi_n)_n$ in $L^{\\Phi^*}$ admits a sequence of forward convex combinations $\\bar{\\xi}_n\\in\\mathrm{conv}(\\xi_n,\\xi_{n+1},...)$ such that $\\sup_n|\\bar{\\xi}_n|\\in L^{\\Phi^*}$ and $\\bar{\\xi}_n$ converges a.s.

Motivation & Objective

• To characterize the lower semicontinuity and continuity of convex functions on the dual space $L^{\Phi^*}$ of a $Δ_2$-Orlicz space $L^\Phi$ with respect to the Mackey topology.
• To establish a connection between topological properties of convex functions and convergence in probability on order intervals in $L^{\Phi^*}$.
• To prove a Komlós-type theorem for norm-bounded sequences in $L^{\Phi^*}$, ensuring almost sure convergence of convex combinations with uniform integrability in the Orlicz norm.
• To provide a functional-analytic framework for studying convex functions on Orlicz dual spaces using probabilistic convergence modes.

Proposed method

• Utilizes the Mackey topology $\tau(L^{\Phi^*}, L^\Phi)$ as the primary topological structure for analyzing convex functions on $L^{\Phi^*}$.
• Applies the concept of order intervals $[-\zeta, \zeta]$ in $L^{\Phi^*}$ to localize the study of lower semicontinuity and continuity.
• Introduces a Komlós-type result: every norm-bounded sequence in $L^{\Phi^*}$ admits a sequence of forward convex combinations $\bar{\xi}_n$ such that $\sup_n |\bar{\xi}_n| \in L^{\Phi^*}$ and $\bar{\xi}_n \to \xi$ almost surely.
• Relies on convergence in probability as a key mode of convergence to characterize topological behavior of convex functions on order intervals.
• Establishes equivalence between Mackey topology continuity and continuity with respect to convergence in probability on order intervals.

Experimental results

Research questions

• RQ1 When is a proper convex function on $L^{\Phi^*}$ lower semicontinuous for the Mackey topology?
• RQ2 When is a finite convex function on $L^{\Phi^*}$ continuous for the Mackey topology?
• RQ3 How can the behavior of convex functions on $L^{\Phi^*}$ be characterized using convergence in probability on order intervals?
• RQ4 What kind of convergence properties can be guaranteed for norm-bounded sequences in $L^{\Phi^*}$ via convex combinations?

Key findings

• A proper convex function on $L^{\Phi^*}$ is lower semicontinuous for the Mackey topology $\tau(L^{\Phi^*}, L^\Phi)$ if and only if it is lower semicontinuous on each order interval $[-\zeta, \zeta]$ with respect to convergence in probability.
• A finite convex function on $L^{\Phi^*}$ is continuous for the Mackey topology if and only if it is continuous on each order interval $[-\zeta, \zeta]$ with respect to convergence in probability.
• Every norm-bounded sequence $(\xi_n)$ in $L^{\Phi^*}$ admits a sequence of forward convex combinations $\bar{\xi}_n \in \mathrm{conv}(\xi_n, \xi_{n+1}, \dots)$ such that $\sup_n |\bar{\xi}_n| \in L^{\Phi^*}$ and $\bar{\xi}_n \to \xi$ almost surely for some $\xi \in L^{\Phi^*}$.
• The uniform integrability condition $\sup_n |\bar{\xi}_n| \in L^{\Phi^*}$ ensures that the limiting behavior of convex combinations is controlled in the Orlicz norm.