Abstract
We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov’s system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes’ theorem. Our probability may take values which are zero-divisors and we discuss carefully this peculiarity.
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Accardi, L.: Quantum Probability: An Introduction to Some Basic Ideas and Trends. Stochastic Models, II. In: Hernandez, D., Lopez-Mimbela, J.A., Quezada, R. (eds.) Aportaciones Mat. Investig., vol. 16. Soc. Mat. Mexicana, Mexico, pp. 1–128 (2001)
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer Briefs in Mathematics, vol. XV (2014)
Luna-Elizarrarás M.E., Perez-Regalado C.O., Shapiro M.: On linear functionals and Hahn–Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebras 23(4), 1105–1129 (2014)
Luna-Elizarrarás M.E., Shapiro M., Struppa D.C., Vajiac A.: Bicomplex numbers and their elementary functions. Cubo A Math. J. 14(2), 61–80 (2012)
Luna-Elizarrarás M.E., Shapiro M., Struppa D.C., Vajiac A.: Complex Laplacian and derivatives of bicomplex functions. Complex Anal. Oper. Theory 7(5), 1675–1711 (2013)
Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers, Frontiers in Mathematics. Birkhäuser, Basel (2015)
Meyer, P.A.: Quantum Probability for Probabilists. Lecture Notes in Mathematics. Springer, Berlin (1993)
Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics, vol. 85. Birkhäuser, Basel (1992)
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M. Elena Luna-Elizarrarás and Michael Shapiro–On leave from Escuela Superior de Fisica y Matemáticas, Instituto Politécnico Nacional.
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Alpay, D., Luna-Elizarrarás, M.E. & Shapiro, M. Kolmogorov’s Axioms for Probabilities with Values in Hyperbolic Numbers. Adv. Appl. Clifford Algebras 27, 913–929 (2017). https://doi.org/10.1007/s00006-016-0706-6
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DOI: https://doi.org/10.1007/s00006-016-0706-6