Tim Sullivan

Cameron-Martin theorems for Cauchy-distributed random sequences

Preprint: Cameron-Martin theorems for Cauchy-distributed random sequences

Han Cheng Lie and I have just uploaded a preprint of our latest paper, on Cameron–Martin-type theorems for sequences of Cauchy-distributed random variables, to the arXiv. Inspired by questions of prior robustness left unanswered in this earlier paper on α-stable Banach space priors, this paper addresses the basic probabilistic question: when is an infinite-dimensional Cauchy distribution, e.g. on sequence space, mutually absolutely continuous with its image under a translation? In the Gaussian case, the celebrated Cameron–Martin theorem says that this equivalence of measures holds if a weighted \(\ell^{2}\) norm (the Cameron–Martin norm) of the translation vector is finite. We show that, in the Cauchy case, the same weighted version of the translation vector needs to lie in the sequence space \(\ell^{1} \cap \ell \log \ell\). More precisely, if the Cauchy distribution on the nth term of the sequence has width parameter \(\gamma_{n} > 0\), and the translation vector is the sequence \(\varepsilon = (\varepsilon_n)_{n = 1}^{\infty}\), then a sufficient condition for mutual absolute continuity is that

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| < \infty \)

and, with the usual convention that \(0 \log 0 = 0\),

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \log \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| \right| < \infty . \)

We also discuss similar results for dilation of the scale parameters, i.e. \(\gamma_{n} \mapsto \sigma_{n} \gamma_{n}\) for some real sequence \(\sigma = (\sigma_n)_{n = 1}^{\infty}\).

Published on Monday 15 August 2016 at 10:00 UTC #publication #preprint