Exact and Asymptotic Distribution Theory for General Gaussian Processes

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This project will further the development of exact and asymptotic distribution theory for Gaussian processes and their quadratic forms. While modern advances in data science owe much progress to computational methods and the rapid growth in computer technology, statistics and applied probability are rife with examples where a careful mathematical analysis allows discoveries that no amount of computational power can uncover. This project is one such example and will use the PI's work on Yule's so called "nonsense" correlation, a 90-year old open problem that was solved last year via mathematical analysis tools. This explicit calculation showed the precise scale of the apparent correlation between two independent continuous series of data, such as what one encounters in economics, climate science, finance, and many other fields. This mathematical explanation of an apparent statistical paradox will enable the investigation of other important questions in mathematical statistics. The project will investigate a possible connection between some important open questions and a set of tools in probability theory whose power mathematical statisticians have only begun to investigate. The project will provide fertile ground for statistics graduate student training at Rice and Michigan State Universities; students will benefit from a wide scope of opportunities, from rigorous study of mathematical tools, to their use in statistics, to applications in fields of great societal value.This project will investigate the probability law of the Pearson correlation between two independent or dependent Gaussian processes. Analyses of distributions in the second Wiener chaos (quadratic forms of normals) are a new set of tools that will be brought to bear. Those tools are flexible enough to handle any Gaussian process via their so-called Karhunen-Loeve expansions. In terms of applications, what is most striking is that any statistical estimation or test based on these projected studies would only require a single or a pair of observations; this is particularly useful for situations, such as in environmental statistics or in economics, where experiments cannot be designed, and one has to work with the available observable data collected dynamically in time. The second emphasis in this study, on Polya frequency functions and related densities, uses some of the same mathematical tools, thanks to a realization that the densities can be represented and expanded explicitly in the second Wiener chaos. The project seeks to prove when a density is strongly log-concave (e.g. its logarithm has a second derivative which is bounded away from zero.) This question, which in mathematical statistics is phrased more broadly in terms of Polya frequency functions, has distribution of sums of independent and non-identically distributed exponentials, expands to the case of general second-chaos distributions. The project could have important consequences in the practice of statistics, especially in areas where comparing non-trivial time series is a challenge, and in many scientific fields informed by properties of log-concavity and strong log-concavity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.