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Large Language Models (LLMs) frequently memorize long sequences verbatim, often with serious legal and privacy implications. Much prior work has studied such verbatim memorization using observational data. To complement such work, we…
Auto- and cross-spectral density functions for dynamic {random} fields and power are derived. These are based on first- and second-order Pad\'{e} approximants of correlation functions expanded in terms of spectral moments. The second-order…
Starting with the Fourier integral theorem, we present natural Monte Carlo estimators of multivariate functions including densities, mixing densities, transition densities, regression functions, and the search for modes of multivariate…
Sparse functional/longitudinal data have attracted widespread interest due to the prevalence of such data in social and life sciences. A prominent scenario where such data are routinely encountered are accelerated longitudinal studies,…
Spectral density estimation is a core problem of system identification, which is an important research area of system control and signal processing. There have been numerous results on the design of spectral density estimators. However to…
In this chapter we take a look at the universal approximation question for stochastic feedforward neural networks. In contrast to deterministic networks, which represent mappings from a set of inputs to a set of outputs, stochastic networks…
If an experimentalist observes a sequence of emitted quantum states via either projective or positive-operator-valued measurements, the outcomes form a time series. Individual time series are realizations of a stochastic process over the…
Neuronal networks provide living organisms with the ability to process information. They are also characterized by abundant recurrent connections, which give rise to strong feedback that dictates their dynamics and endows them with fading…
In the univariate setting, using the kernel spectral representation is an appealing approach for generating stationary covariance functions. However, performing the same task for multiple-output Gaussian processes is substantially more…
Memory consistency models define the order in which accesses to shared memory in a concurrent system may be observed to occur. Such models are a necessity since program order is not a reliable indicator of execution order, due to…
We study the asymptotic behavior of wavelet coefficients of random processes with long memory. These processes may be stationary or not and are obtained as the output of non--linear filter with Gaussian input. The wavelet coefficients that…
This paper describes limiting behaviour of tail empirical process associated with long memory stochastic volatility models. We show that such process has dichotomous behaviour, according to an interplay between a Hurst parameter and a tail…
The spectrum and coherency are useful quantities for characterizing the temporal correlations and functional relations within and between point processes. This paper begins with a review of these quantities, their interpretation and how…
A density matrix describes the statistical state of a quantum system. It is a powerful formalism to represent both the quantum and classical uncertainty of quantum systems and to express different statistical operations such as measurement,…
A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as…
This paper proposes an approach to the joint modeling of the short-time Fourier transform magnitude and phase spectrograms with a deep generative model. We assume that the magnitude follows a Gaussian distribution and the phase follows a…
Multiprocess systems, including grid systems, multiprocessors and multicore computers, incorporate a variety of specialized hardware and software mechanisms, which speed computation, but result in complex memory behavior. As a consequence,…
Layered stable (multivariate) distributions and processes are defined and studied. A layered stable process combines stable trends of two different indices, one of them possibly Gaussian. More precisely, in short time, it is close to a…
Memory consistency models have been developed to specify what values may be returned by a read given that, in a distributed system, memory operations may only be partially ordered. Before this work, consistency models were defined…
Stochastic chains represent a wide and key variety of phenomena in many branches of science within the context of Information Theory and Thermodynamics. They are typically approached by a sequence of independent events or by a memoryless…