arXiv subjects
Publications and source records attributed to David G Maher.
This paper uses the wrapping map of Dooley and Wildberger to prove Lp boundedness of multipliers on compact Lie groups by transferring the estimate from Rn. This improves the bounds in several cases, and simplifies the proofs of others.
Stress shocks are often calculated as multiples of the standard deviation of a history set. This paper investigates how many standard deviations are required to guarantee that this shock exceeds any observation within the history set, given the additional constraint of kurtosis. The results of this analysis are then used to validate the shocks produced by some stress test models, in particular that of Brace-Lauer-Rado. A secondary application of our results is to investigate three known extensions of Chebyshev's Inequality where the kurtosis is known. It is found that our results give a tighter bound than the well-known inequalities.
An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements also hold in the case of a Lie group. By using the wrapping map of Dooley and Wildberger, we show how to wrap a Brownian motion to a compact Lie group from its Lie algebra (viewed as a Euclidean space) and find the heat kernel. This is achieved by considering Itô type stochastic differential equations and applying the Feynman-Kač theorem.
In this paper we extend our previous results on wrapping Brownian motion and heat kernels onto compact Lie groups to various symmetric spaces, where a global generalisation of Rouvière's formula and the $e$-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.
We examine three methods of constructing correlated Student-$t$ random variables. Our motivation arises from simulations that utilise heavy-tailed distributions for the purposes of stress testing and economic capital calculations for financial institutions. We make several observations regarding the suitability of the three methods for this purpose.