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Journal articles on the topic 'Probability Density Functionals'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Joachim, C. "The probability and energy density currents as density functionals." Journal of Physics A: Mathematical and General 19, no. 13 (1986): 2549–57. http://dx.doi.org/10.1088/0305-4470/19/13/020.

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2

Lapko, Aleksandr V., and Vasiliy A. Lapko. "Estimation of the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables." Metrologiya, no. 1 (2020): 15–27. http://dx.doi.org/10.32446/0132-4713.2020-1-15-27.

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When substantiating the method of fast selection of the bandwidth of kernel probability density estimates, a constant was found that is a functional of the second density derivative. In this paper, the obtained result is generalized to derivatives of symmetric probability densities of different orders. The functional dependences of the constants under study on the coeffi cient of antikurtosis of a random variable are established. The regularities peculiar to them are investigated. Based on the results obtained, a method for estimating functionals from derived probability densities has been dev

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3

Gr�bel, Rudolf. "Estimation of density functionals." Annals of the Institute of Statistical Mathematics 46, no. 1 (1994): 67–75. http://dx.doi.org/10.1007/bf00773593.

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4

Alonso, Andrés M., and Antonio Cuevas. "On smoothed bootstrap for density functionals." Journal of Nonparametric Statistics 15, no. 4-5 (2003): 467–77. http://dx.doi.org/10.1080/10485250310001604613.

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5

Borovikov, V. P. "Estimation of Functionals of a Density." Theory of Probability & Its Applications 37, no. 3 (1993): 507–14. http://dx.doi.org/10.1137/1137100.

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6

Nadaraya, Elizbar, and Grigol Sokhadze. "On integral functionals of a density." Communications in Statistics - Theory and Methods 45, no. 23 (2016): 7086–102. http://dx.doi.org/10.1080/03610926.2014.974825.

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7

Lapko, Aleksandr V., and Vasiliy A. Lapko. "Estimation of a nonlinear functional from the probability density when optimizing nonparametric decision functions." Izmeritel`naya Tekhnika, no. 1 (2021): 14–20. http://dx.doi.org/10.32446/0368-1025it.2021-1-14-20.

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A method for estimating the nonlinear functional of the probability density of a two-dimensional random variable is proposed. It is relevant when implementing procedures for fast bandwidths selection in the problem of optimization of kernel probability density estimates. The solution of this problem allows to significantly improve the computational efficiency of nonparametric decision rules. The basis of the proposed approach is the analysis of the formula for the optimal bandwidth of the kernel probability density estimation. In this case, the bandwidth of kernel functions is represented as t

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8

Martinez, Harú V., and Marı́a M. Olivares. "Estimation of quadratic functionals of a density." Statistics & Probability Letters 42, no. 4 (1999): 327–32. http://dx.doi.org/10.1016/s0167-7152(98)00193-x.

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9

BURKE, KIERON, and JOHN P. PERDEW. "DENSITY FUNCTIONALS AND SMALL INTERPARTICLE SEPARATIONS IN ELECTRONIC SYSTEMS." Modern Physics Letters B 09, no. 14 (1995): 829–38. http://dx.doi.org/10.1142/s0217984995000784.

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We review some recent results concerning the probability that two electrons will be found close together in any interacting electronic system, and why this probability is usually well approximated by local (LSD) and semilocal spin density functional theories. The success of these approximations for the energy in "normal" systems is explained by the usual sum rule arguments on the system- and spherically-averaged exchange-correlation hole density <n xc (u)>, coupled with the nearly correct, but not exact, behavior of these approximations as the interelectronic separation u → 0. We argue t

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10

Donoho, David L. "One-Sided Inference about Functionals of a Density." Annals of Statistics 16, no. 4 (1988): 1390–420. http://dx.doi.org/10.1214/aos/1176351045.

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11

Lindstrom, Michael R., Hyuntae Jung, and Denis Larocque. "Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection." Entropy 22, no. 12 (2020): 1363. http://dx.doi.org/10.3390/e22121363.

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We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a “point approach”), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a “Fourier app

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12

Pardo, J. C., V. Rivero, and K. van Schaik. "On the density of exponential functionals of Lévy processes." Bernoulli 19, no. 5A (2013): 1938–64. http://dx.doi.org/10.3150/12-bej436.

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13

Martinucci, Barbara, and Alessandra Meoli. "Certain functionals of squared telegraph processes." Stochastics and Dynamics 20, no. 01 (2019): 2050005. http://dx.doi.org/10.1142/s0219493720500057.

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We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment generating function. Some results on the first-passage time through a fixed positive level are also provided. Moreover, we analyze some functionals [Formula: see text] of two independent squared telegraph processes, both in the case [Formula: see text] and [Formula: see text]. Starting from this study, we provide some results on the probability density functions of the two-dimensional radial telegraph

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14

Gittenberger, Bernhard, and Guy Louchard. "The Brownian excursion multi-dimensional local time density." Journal of Applied Probability 36, no. 2 (1999): 350–73. http://dx.doi.org/10.1239/jap/1032374458.

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Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: a direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for Galton–Watson trees.

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15

Gittenberger, Bernhard, and Guy Louchard. "The Brownian excursion multi-dimensional local time density." Journal of Applied Probability 36, no. 02 (1999): 350–73. http://dx.doi.org/10.1017/s0021900200017186.

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Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: a direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for Galton–Watson trees.

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16

Penskaya, M. Ya. "Estimation of functionals in an a priori density." Journal of Soviet Mathematics 41, no. 1 (1988): 841–46. http://dx.doi.org/10.1007/bf01097898.

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17

GINÉ, EVARIST, and DAVID M. MASON. "Uniform in Bandwidth Estimation of Integral Functionals of the Density Function." Scandinavian Journal of Statistics 35, no. 4 (2008): 739–61. http://dx.doi.org/10.1111/j.1467-9469.2008.00600.x.

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18

Hasminskii, R. Z., and I. A. Ibragimov. "Asymptotically efficient nonparametric estimation of functionals of a spectral density function." Probability Theory and Related Fields 73, no. 3 (1986): 447–61. http://dx.doi.org/10.1007/bf00776242.

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19

Ginovyan, M. S. "Asymptotically Efficient Nonparametric Estimation of Functionals of a Spectral Density Having Zeros." Theory of Probability & Its Applications 33, no. 2 (1989): 296–303. http://dx.doi.org/10.1137/1133045.

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20

Chen, Su. "A new distribution‐free k ‐sample test: Analysis of kernel density functionals." Canadian Journal of Statistics 48, no. 2 (2019): 167–86. http://dx.doi.org/10.1002/cjs.11525.

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21

Ludeña, Carenne. "Efficient estimation of functionals of the spectral density of stationary Gaussian fields." ESAIM: Probability and Statistics 3 (1999): 23–47. http://dx.doi.org/10.1051/ps:1999101.

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22

Wu, Tiee-Jian, Chih-Yuan Hsu, Huang-Yu Chen, and Hui-Chun Yu. "Root $$n$$ n estimates of vectors of integrated density partial derivative functionals." Annals of the Institute of Statistical Mathematics 66, no. 5 (2013): 865–95. http://dx.doi.org/10.1007/s10463-013-0428-7.

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23

Silva, Luís M., J. Marques de Sá, and Luís A. Alexandre. "The MEE Principle in Data Classification: A Perceptron-Based Analysis." Neural Computation 22, no. 10 (2010): 2698–728. http://dx.doi.org/10.1162/neco_a_00013.

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This letter focuses on the issue of whether risk functionals derived from information-theoretic principles, such as Shannon or Rényi's entropies, are able to cope with the data classification problem in both the sense of attaining the risk functional minimum and implying the minimum probability of error allowed by the family of functions implemented by the classifier, here denoted by min Pe. The analysis of this so-called minimization of error entropy (MEE) principle is carried out in a single perceptron with continuous activation functions, yielding continuous error distributions. In spite of

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24

Berzin, Corinne, José León, and Joaquín Ortega. "Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates." Statistics 37, no. 3 (2003): 217–42. http://dx.doi.org/10.1080/02331880290015440.

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25

Chacón, José E., and Carlos Tenreiro. "Exact and Asymptotically Optimal Bandwidths for Kernel Estimation of Density Functionals." Methodology and Computing in Applied Probability 14, no. 3 (2011): 523–48. http://dx.doi.org/10.1007/s11009-011-9243-x.

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26

Ayanbayev, Birzhan, Ilja Klebanov, Han Cheng Lie та T. J. Sullivan. "Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces". Inverse Problems 38, № 2 (2021): 025006. http://dx.doi.org/10.1088/1361-6420/ac3f82.

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Abstract We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures

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27

Frikha, Noufel, and Libo Li. "Weak uniqueness and density estimates for SDEs with coefficients depending on some path-functionals." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 56, no. 2 (2020): 1002–40. http://dx.doi.org/10.1214/19-aihp992.

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28

Park, Hyun-Suk. "Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions." Mathematics 13, no. 2 (2025): 323. https://doi.org/10.3390/math13020323.

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Let G be a random variable of functionals of an isonormal Gaussian process X defined on some probability space. Studies have been conducted to determine the exact form of the density function of the random variable G. In this paper, unlike previous studies, we will use the Stein’s method for invariant measures of diffusions to obtain the density formula of G. By comparing the density function obtained in this paper with that of the diffusion invariant measure, we find that the diffusion coefficient of an Itô diffusion with an invariant measure having a density can be expressed as in terms of o

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29

Gittenberger, Bernhard, and Guy Louchard. "On the local time density of the reflecting Brownian bridge." Journal of Applied Mathematics and Stochastic Analysis 13, no. 2 (2000): 125–36. http://dx.doi.org/10.1155/s1048953300000149.

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Expressions for the multi-dimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings.

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30

Dokuchaev, N. G. "Distributions of Itô Processes: Estimates for the Density and for Conditional Expectations of Integral Functionals." Theory of Probability & Its Applications 39, no. 4 (1995): 662–70. http://dx.doi.org/10.1137/1139051.

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31

Hürlimann, Werner. "Analytical Evaluation of Economic Risk Capital for Portfolios of Gamma Risks." ASTIN Bulletin 31, no. 1 (2001): 107–22. http://dx.doi.org/10.2143/ast.31.1.996.

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AbstractBased on the notions of value-at-risk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). Quantitatively, RaC is equal to VaR plus an additional stop-loss dependent term, which takes into account the average amount at loss. Furthermore, RaC is additi

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32

Mukherjee, Rajarshi, and Bodhisattva Sen. "On efficiency of the plug-in principle for estimating smooth integrated functionals of a nonincreasing density." Electronic Journal of Statistics 13, no. 2 (2019): 4416–48. http://dx.doi.org/10.1214/19-ejs1629.

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33

Kendall, W. S., M. N. M. van Lieshout, and A. J. Baddeley. "Quermass-interaction processes: conditions for stability." Advances in Applied Probability 31, no. 02 (1999): 315–42. http://dx.doi.org/10.1017/s0001867800009137.

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We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components m

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34

Sharma, Akariti. "On non-monotonic temperature dependence of linear contact probability in atomic chains." Physica Scripta 96, no. 12 (2021): 125723. http://dx.doi.org/10.1088/1402-4896/ac387a.

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Abstract In this paper, we theoretically investigate the temperature T dependence of linear contact probability in atomic chains. In these chains, the transverse motion of the electrons is assumed to be confined in the harmonic oscillator confinement model. The intra-chain electron correlations are treated within both Hartree–Fock Approximation (HFA) and Random Phase Approximation (RPA). As artifact HFA correlation functions are used in the RPA perturbative calculations. Results are presented for numerically computed pair-correlation functions at different temperatures and electron densities.

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35

Ishiyama, Kazuyuki. "Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion." Methodology and Computing in Applied Probability 7, no. 3 (2005): 271–83. http://dx.doi.org/10.1007/s11009-005-4517-9.

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36

Luz, Maksym, and Mikhail Moklyachuk. "Estimation problem for continuous time stochastic processes with periodically correlated increments." Statistics, Optimization & Information Computing 11, no. 4 (2023): 811–28. http://dx.doi.org/10.19139/soic-2310-5070-1792.

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We deal with the problem of optimal estimation of the linear functionals constructed from unobserved values of a continuous time stochastic process with periodically correlated increments based on past observations of this process. To solve the problem, we construct a corresponding to the process sequence of stochastic functions which forms an infinite dimensional vector stationary increment sequence. In the case of known spectral density of the stationary increment sequence, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal est

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37

Eremina, Tatyana, Johan Debayle, Frederic Gruy, and Jean-Charles Pinoli. "Local Measures Distribution for the Estimation of the Elongation Ratio of the Typical Grain in Homogeneous Boolean Models." Image Analysis & Stereology 40, no. 2 (2021): 95–103. http://dx.doi.org/10.5566/ias.2554.

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We introduce a particular localization of the Minkowski functionals to characterize and discriminate different random spatial structures. The aim of this paper is to present a method estimating the typical grain elongation ratio in a homogeneous Boolean model. The use of this method is demonstrated on a range of Boolean models of rectangles featuring fixed and random elongation ratio. An optimization algorithm is performed to determine the elongation ratio which maximize the likelihood function of the probability density associated with the local perimeter measure. Therefore, the elongation ra

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38

Heydenreich, Markus, Remco van der Hofstad, and Georgi Radulov. "FUNCTIONALS OF BROWNIAN BRIDGES ARISING IN THE CURRENT MISMATCH IN D/A CONVERTERS." Probability in the Engineering and Informational Sciences 23, no. 1 (2008): 149–72. http://dx.doi.org/10.1017/s0269964809000114.

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Digital-to-analog converters (DAC) transform signals from the abstract digital domain to the real analog world. In many applications, DACs play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Nonlinearity (INL), which we study in this article. There are several DAC architectures. The most widely used architectures are the thermometer and the bi

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39

Chacón, José E., and Tarn Duong. "Efficient recursive algorithms for functionals based on higher order derivatives of the multivariate Gaussian density." Statistics and Computing 25, no. 5 (2014): 959–74. http://dx.doi.org/10.1007/s11222-014-9465-1.

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40

Pattnaik, Jeet Amrit, Joshua T. Majekodunmi, Mrutunjaya Bhuyan, and Suresh Kumar Patra. "Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional." Foundations 2, no. 4 (2022): 898–911. http://dx.doi.org/10.3390/foundations2040061.

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The present study is focused on revealing a characteristic kink of the neutron shell closure N = 126 across the Hg-isotopic chain within the relativistic mean-field (RMF) approach with the IOPB-I, DD-ME2, DD-PC1 and NL3 parameter sets. The RMF densities are converted to their spherical equivalence via the Wood–Saxon approximation and used as input within the parametrization procedure of the coherent density fluctuation model (CDFM). The nuclear matter symmetry energy is calculated using the Brückner energy density functional, and its surface, as well as volume components, are evaluated within

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41

Rastegin, Alexey E. "Number-phase uncertainty relations in terms of generalized entropies." Quantum Information and Computation 12, no. 9&10 (2012): 743–62. http://dx.doi.org/10.26421/qic12.9-10-2.

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Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequa

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42

Dehesa, Jesús S. "Entropy-Like Properties and Lq-Norms of Hypergeometric Orthogonal Polynomials: Degree Asymptotics." Symmetry 13, no. 8 (2021): 1416. http://dx.doi.org/10.3390/sym13081416.

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In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the Lq

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43

Rolski, Tomasz. "A NOTE ON THE INCREASING DIRECTIONALLY CONCAVE MONOTONICITY IN QUEUES." Probability in the Engineering and Informational Sciences 19, no. 1 (2005): 33–43. http://dx.doi.org/10.1017/s0269964805050035.

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In this article, we study comparison theorems for stochastic functionals like V(∞;C) = sup0≤t {M(t) − C(t)} or V(T;C) = sup0≤t≤T {M(t) − C(t)}, where {M(t)} and {C(t)} are two independent nondecreasing processes with stationary increments. We will tacitly assume that the considered stochastic functionals are proper random variables. We prove that V(T;C′) ≤icxV(T;C) ≤icxV(T;C′′), where and C′′(dt) = c(0) dt, provided dC(t) is absolute continuous with density c(t). Similarly, we show that V(∞;C′) ≤icxV(∞;C) ≤icxV(∞;C′′). For proofs, we develop the theory of the ≤idcv ordering defined by increasi

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44

Kuznetsov, A. "Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions." Journal of Applied Probability 47, no. 4 (2010): 1023–33. http://dx.doi.org/10.1239/jap/1294170516.

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In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density o

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45

Debbi, Latifa. "Explicit solutions of some fractional partial differential equations via stable subordinators." Journal of Applied Mathematics and Stochastic Analysis 2006 (February 26, 2006): 1–18. http://dx.doi.org/10.1155/jamsa/2006/93502.

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The aim of this work is to represent the solutions of one-dimensional fractional partial differential equations (FPDEs) of order (α∈ℝ+\ℕ) in both quasi-probabilistic and probabilistic ways. The canonical processes used are generalizations of stable Lévy processes. The fundamental solutions of the fractional equations are given as functionals of stable subordinators. The functions used generalize the functions given by the Airy integral of Sirovich (1971). As a consequence of this representation, an explicit form is given to the density of the 3/2-stable law and to the density of escaping islan

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46

Gautier, Athénaïs, David Ginsbourger, and Guillaume Pirot. "Goal-oriented adaptive sampling under random field modelling of response probability distributions." ESAIM: Proceedings and Surveys 71 (August 2021): 89–100. http://dx.doi.org/10.1051/proc/202171108.

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In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, a

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47

Roy, S., A. Borzì, and A. Habbal. "Pedestrian motion modelled by Fokker–Planck Nash games." Royal Society Open Science 4, no. 9 (2017): 170648. http://dx.doi.org/10.1098/rsos.170648.

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A new approach to modelling pedestrians' avoidance dynamics based on a Fokker–Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal cont

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48

Squicciarini, Antonio, Elio Valero Toranzo, and Alejandro Zarzo. "A Time-Series Feature-Extraction Methodology Based on Multiscale Overlapping Windows, Adaptive KDE, and Continuous Entropic and Information Functionals." Mathematics 12, no. 15 (2024): 2396. http://dx.doi.org/10.3390/math12152396.

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We propose a new methodology to transform a time series into an ordered sequence of any entropic and information functionals, providing a novel tool for data analysis. To achieve this, a new algorithm has been designed to optimize the Probability Density Function (PDF) associated with a time signal in the context of non-parametric Kernel Density Estimation (KDE). We illustrate the applicability of this method for anomaly detection in time signals. Specifically, our approach combines a non-parametric kernel density estimator with overlapping windows of various scales. Regarding the parameters i

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49

Bressloff, Paul C. "Diffusion-mediated absorption by partially-reactive targets: Brownian functionals and generalized propagators." Journal of Physics A: Mathematical and Theoretical 55, no. 20 (2022): 205001. http://dx.doi.org/10.1088/1751-8121/ac5e75.

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Abstract Many processes in cell biology involve diffusion in a domain Ω that contains a target U whose boundary ∂ U is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on

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50

Ellinas, Demosthenes, and Ioannis Tsohantjis. "Random Walk and Diffusion on a Smash Line Algebra." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 02 (2003): 245–64. http://dx.doi.org/10.1142/s0219025703001158.

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Abstract:

Working within the framework of Hopf algebras, a random walk and the associated diffusion equation are constructed on a space that is algebraically described as the merging of the real line algebra with the anyonic line algebra. Technically this merged structure is a smash algebra, namely an algebra resulting from a braided tensoring of real with anyonic line algebras. The motivation of introducing the smashing results from the necessity of having noncommuting increments in the random walk. Based on the observable-state duality provided by the underlying Hopf structure, the construction is cas

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