Independence arising from interacting Fock spaces and related central limit theorem

• Autorzy: Ben Ghorbal, A. , Crismale, V.
• Języki publikacji: EN

Abstrakty

EN

We present the notion of projective independence, which abstracts, in an algebraic setting, the factorization rule for the vacuum expectation of creation-annihilations-preservation operators in interacting Fock spaces described in [3]. Furthermore, we give a central limit theorem based on such a notion and a Fock representation of the limit process.

Słowa kluczowe

EN

interacting Fock spaces; central limit theorems; GNS representation

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2009
• Tom: Vol. 29, Fasc. 2
• Strony: 251--269
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Ben Ghorbal, A.

• Department of Mathematics, Faculty of Sciences, Al-Imam Muhammad Ibn Saud Islamic University, P. O. Box 90050, 11623 Riyadh, Kingdom of Saudi Arabia,

autor

Crismale, V.

• Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona, 4-70125 Bari, Italy,

Bibliografia

• [1] L. Accardi and A. Bach, The harmonic oscillator as quantum central limit theorem for Bernoulli processes, preprint, 1985.
• [2] L. Accardi and M. Bożejko, Interacting Fock spaces and gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), pp. 663-670.
• [3] L. Accardi, V. Crismale and Y. G. Lu, Universal central limit theorems based on interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), pp. 631-650.
• [4] L. Accardi, A. Frigerio and J. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18 (1982), pp. 97-133.
• [5] L. Accardi, Y. Hashimoto and N. Obata, Notions of independence related to the free group, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), pp. 201-220.
• [6] L. Accardi, Y. G. Lu and I. Volovich, The QED Hilbert module and interacting Fock spaces, International Institute for Advances Studies, Kyoto 1997.
• [7] A. Ben Ghorbal, V. Crismale and Y. G. Lu, A constructive boolean central limit theorem, Boll. Unione Mat. Ital. Sez. B 10 (2007), pp. 593-604.
• [8] A. Ben Ghorbal and M. Schürmann, Non-commutative notions of stochastic independence, Math. Proc. Cambridge Philos. Soc. 133 (2002), pp. 531-561.
• [9] M. Bożejko and R. Speicher, Interpolations between bosonic and fermionic relations given by generalized Brownian motions, Math. Z. 222 (1996), pp. 135-159.
• [10] V. Crismale, A projective central limit theorem and interacting Fock space representation for the limit process, Banach Center Publ. 78 (2008), pp. 69-80.
• [11] M. De Giosa and Y. G. Lu, The free creation and annihilation operators as the central limit of the quantum Bernoulli process, Random Oper. Stochastic Equations 5 (1997), pp. 227-236.
• [12] N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, Z. Wahrsch. Verw. Gebiete 42 (1978), pp. 129-134.
• [13] A. Krystek and L. Wojakowski, Convolution and central limit theorem arising from addition of field operators in one mode type interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), pp. 651-658.
• [14] V. Liebsher, On a central limit theorem for monotone noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), pp. 155-167.
• [15] N. Muraki, Monotone independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), pp. 39-58.
• [16] N. Muraki, The five independencies as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), pp. 333-371.
• [17] M. Schürmann, Non-commutative probability on algebraic structures, in: Probability Measures on Groups and Related Structures XI, World Scientific, River Edge, NJ, 1995, pp. 332-356.
• [18] R. Speicher, On universal product, Free Probab. Th., Fields Inst. Commun. 12 (1997), pp. 257-266.
• [19] R. Speicher and W. von Waldenfels, A general central limit theorem and invariance principle, in: Quantum Probability and Related Topics IX, L. Accardi (Ed.), World Scientific, River Edge, NJ, 1994, pp. 371-387.