A detailed mathematical investigation of multibit quantizing systems using nonsubtractive dither is presented. It is shown that by the use of dither having a suitably chosen probability density function, moments of the total error can be made independent of the system input signal but that statistical independence of the error and the input signals is not achievable. Similarly, it is demonstrated that values of the total error signal cannot generally be rendered statistically independent of one another but that their joint moments can be controlled and that, in particular, the error sequence can be rendered spectrally white. The properties of some practical dither signals are explored, and recommendations are made for dithering in audio, video, and measurement applications. The paper collects all of the important results on the subject of nonsubtractive dithering and introduces important new ones with the goal of alleviating persistent and widespread misunderstandings regarding the technique.

A theory of nonsubtractive dither

Analogues of the Marcinkiewicz-Zygmund and Rosenthal inequalities for Banach space valued random vectors are proved. As an application some results on the strong law of large numbers are obtained. It is proved that the Marcinkiewicz SLLN holds for every $p$-integrable, mean zero $B$-valued $\mathrm{rv}$ if and only if $B$ is of Rademacher type $p(1 \leq p < 2)$.

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Inequalities for b-valued random vectors with applications to the strong law of large numbers

We consider asymptotic properties of the least squares estimator (LSE) in a regression model with long-memory stationary errors. First we derive a necessary and sufficient condition that the LSE be asymptotically efficient relative to the best linear unbiased estimator (BLUE). Then we derive the asymptotic distribution of the LSE under a condition on the higher-order cumulants of the white-noise process of the errors.

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Asymptotic Properties of the LSE in a Regression Model with Long-Memory Stationary Errors

Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.

Nonclassical moments and their measurement

Exponential functionals of Levy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping $$\Phi $$
 , which maps two independent Levy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Levy) characteristics. Also, continuity of $$\Phi $$
 is treated, and some results on its range are obtained.

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Tatsuo Kawata

Papers

Fourier analysis in probability theory