arXiv subjects

Bruce Alastair Watson

Publications and source records attributed to Bruce Alastair Watson.

11 recordsLinked to original sources

arXiv2025

A Kakutani-Rokhlin decomposition for conditionally ergodic process in the measure-free setting of vector lattices

Recently the Kac formula for the conditional expectation of the first recurrence time of a conditionally ergodic conditional expectation preserving system was established in the measure free setting of vector lattices (Riesz spaces). We now give a formulation of the Kakutani-Rokhlin decomposition for conditionally ergodic systems in terms of components of weak order units in a vector lattice. In addition, we prove that every aperiodic conditional expectation preserving system can be approximated by a periodic system.

math.DS↗

arXiv2022

A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

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.

math.FA↗

arXiv2018

Near-Epoch Dependence in Riesz Spaces

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.

math.FA↗

arXiv2017

Mixing inequalities in Riesz spaces

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.

math.FA↗

arXiv2017

Bernoulli Processes in Riesz spaces

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.

math.FA↗

arXiv2017

Borg's Periodicity Theorems for first order self-adjoint systems with complex potentials

A self-adjoint first order system with Hermitian $π$-periodic potential $Q(z)$, integrable on compact sets, is considered. It is shown that all zeros of $Δ+ 2e^{-i\int_0^π\Im q dt}$ are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which $Q(z)$ is $\fracπ{2}$-periodic. Furthermore, the zeros of $Δ- 2e^{-i\int_0^π\Im q dt}$ are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which $Q(z) = σ_2 Q(z) σ_2$. Here $Δ$ denotes the discriminant of the system and $σ_0$, $σ_2$ are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if $Q = rσ_0 + qσ_2$, for some real valued $π$-periodic functions $r$ and $q$ integrable on compact sets.

math.SP↗

arXiv2017

Indefinite boundary value problems on graphs

We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This leads to a max-min principle and as a consequence we can formulate an analogue of Dirichlet-Neumann bracketing and this in turn gives rise to asymptotic approximations for the eigenvalues.

math.SP↗

arXiv2017

Canonical systems in $\mathbb{R}^2$ with periodic potentials and vanishing instability intervals

Canonical systems in $\mathbb{R}^2$ with absolutely continuous real symmetric $π$-periodic potentials matrices are considered. A through analysis of the discriminant is given along with the indexing and interlacing of the eigenvalues of the periodic, anti-periodic and Dirichlet-type boundary value problems on $[0,π]$. The periodic and anti-periodic eigenvalues are characterized in terms of Dirichlet type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real valued function with the identity matrix.

math.SP↗

arXiv2017

Inverse scattering on the line with a transfer condition

The inverse scattering problem for Sturm-Liouville operators on the line with a matrix transfer condition at the origin is considered. We show that the transfer matrix can be reconstructed from the eigenvalues and reflection coefficient. In addition, for potentials with compact essential support, we show that the potential can be uniquely reconstructed.

math.SP↗

arXiv2017

Markov processes on Riesz spaces

Measure-free discrete time stochastic processes in Riesz spaces were formulated and studied by Kuo, Labuschagne and Watson. Aspects relating martingales, stopping times, convergence of these processes as well as various decomposition were considered. Here we formulate and study Markov processes in a measure-free Riesz space setting.

math.PR↗

arXiv2017

Mixingales on Riesz spaces

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.

math.PR↗