Victor M. Panaretos

Second-Order Comparison of Gaussian Random Functions and the Geometry of DNA Minicircles

Given two samples of continuous zero-mean iid Gaussian processes on [0,1], we consider the problem of testing whether they share the same covariance structure. Our study is motivated by the problem of determining whether the mechanical properties of short strands of DNA are significantly affected by their base-pair sequence; though expected to be true, had so far not been observed in three-dimensional electron microscopy data. The testing problem is seen to involve aspects of ill-posed inverse problems and a test based on a Karhunen–Loève approximation of the Hilbert–Schmidt distance of the empirical covariance operators is proposed and investigated. When applied to a dataset of DNA minicircles obtained through the electron microscope, our test seems to suggest potential sequence effects on DNA shape. Supplemental material available online.

Fourier analysis of stationary time series in function space

We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density operator, which generalises the notion of a spectral density matrix to the functional setting, and characterises the second-order dynamics of the process. Our main tool is the functional Discrete Fourier Transform (fDFT). We derive an asymptotic Gaussian representation of the fDFT, thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex-valued Gaussian random functions. Our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel, the functional generalisation of the periodogram matrix. The consistency and asymptotic law of these estimators are studied in detail. As immediate consequences, we obtain central limit theorems for the mean and the long-run covariance operator of a stationary functional time series. Our results do not depend on structural modelling assumptions, but only functional versions of classical cumulant mixing conditions, and are shown to be stable under discrete observation of the individual curves.

Amplitude and phase variation of point processes

We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function

On Random Tomography with Unobservable Projection Angles

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Asymptotic inference for partially observed branching processes

corresponds to its random oscillations in the

Nonparametric Construction of Multivariate Kernels

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Representation of Radon Shape Diffusions via Hyperspherical Brownian Motion

-axis, typically encapsulated by its (co)variation around a mean level. In contrast, its phase variation refers to fluctuations in the

Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics

x

Sparse Approximations Of Protein Structure From Noisy Random Projections

-axis, often caused by random time changes. We formalise similar notions for a point process, and nonparametrically separate them based on realisations of i.i.d. copies

A Statistician’s View on Deconvolution and Unfolding

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