arXiv subjects
Publications and source records attributed to Wen-Chi Kuo.
For a conditional expectation operator $T$ on a Dedekind complete Riesz space, we give representations of the $T$-strong duals of $L^1(T)$ and $L^\infty(T)$. The representation for the $T$-strong dual of $L^1(T)$ follows from the known result for $L^2(T)$. To describe the $T$-strong dual of $L^\infty(T)$, we introduce charges on components of weak order units and develop a corresponding integration theory.
Martingales, Markov processes and Laws of Large Numbers have been well studied in the Riesz space (vector lattice) setting. There has, however, been no attention given in the Riesz space setting to Laws of Small Numbers or to the so called Stein-Chen method. Here we adapt the Stein-Chen method to the Riesz space setting and hence give a conditional Laws of Small Numbers for Bernoulli processes in Riesz spaces. This requires extensive use of functional calculus and the associated f-algebra structure.
We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An Andô-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a Riesz-Frechet representation theorem for the $T$-strong dual, where $T$ is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required.
We extend the Koopman-von Neumann convergence condition on the Cesàro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in $L^1$.
The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Riesz space setting.
In this paper we generalize the Hájek-Rényi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for $1\le p\le 2$.
The abstraction of the study of stochastic processes to Banach lattices and vector lattices has received much attention by Grobler, Kuo, Labuschagne, Stoica, Troitsky and Watson over the past fifteen years. By contrast mixing processes have received very little attention. In particular mixingales were generalized to the Riesz space setting in {\sc W.-C. Kuo, J.J. Vardy, B.A. Watson,} Mixingales on Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{402} (2013), 731-738. The concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in {\sc W.-C. Kuo, M.J. Rogans, B.A. Watson,} Mixing inequalities in Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{456} (2017), 992-1004. In the present work we formulate the concept of near-epoch dependence for Riesz space processes and show that if a process is near-epoch dependent and either strong or uniform mixing then the process is a mixingale, giving access to a law of large numbers. The above is applied to autoregessive processes of order 1 in Riesz spaces.
Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015), 647-657}}] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$, were discussed in the setting of Riesz spaces as $\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson}, {Discrete stochastic integration in Riesz spaces}, {\em Positivity}, {\bf 14} {(2010), 859-875}}]. An $R(T)$ valued norm, for the cases of $p=1,\infty,$ was introduced on these spaces in [{{\sc W. Kuo, M. Rogans, B.A. Watson}, {Mixing processes in Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 456} {(2017), 992-1004}}] where it was also shown that $R(T)$ is a universally complete $f$-algebra and that these spaces are $R(T)$-modules. In [{{\sc Y. Azouzi, M. Trabelsi}, {$L^p$-spaces with respect to conditional expectation on Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 447} {(2017), 798-816}}] functional calculus was used to consider $\mathcal{L}^{p}(T)$ for $p\in (1,\infty)$. In this paper we prove the strong sequential completeness of the space $\mathcal{L}^{1}(T)$, the natural domain of the conditional expectation operator $T$, and the strong completeness of $\mathcal{L}^{\infty}(T)$.
Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the $α$ (strong) and $φ$ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized $L^1$ and $L^\infty$ spaces begun by Kuo, Labuschagne and Watson will be extended.
The action and averaging properties of conditional expectation operators are studied in the, measure-free, Riesz space, setting of Kuo, Labuschagne and Watson [{Conditional expectations on Riesz spaces}, J. Math. Anal. Appl., 303 (2005), 509-521] but on the abstract $L^2$ space, ${\cal L}^2(T)$ introduced by Labuschagne and Watson [{ Discrete Stochastic Integration in Riesz Spaces}, Positivity, 14, (2010), 859 - 575]. In this setting it is shown that conditional expectation operators leave ${\cal L}^2(T)$ invariant and the Bienaymé equality and Tchebichev inequality are proved. From this foundation Bernoulli processes are considered. Bernoulli's strong law of large numbers and Poisson's theorem are formulated and proved.
A mixingale is a stochastic process which combines properties of martingales and mixing sequences. McLeish introduced the term mixingale at the $4^{th}$ Conference on Stochastic Processes and Application, at York University, Toronto, 1974, in the context of $L^2$. In this paper we generalize the concept of a mixingale to the measure-free Riesz space setting (this generalizes all of the $L^p, 1\le p\le \infty$ variants) and prove that a weak law of large numbers holds for Riesz space mixingales. In the process we also generalize the concept of uniform integrability to the Riesz space setting.