The Karlin-McGregor formula for paths connected with a clique

• Autorzy: Obata N.
• Języki publikacji: EN

Abstrakty

EN

The Karlin-McGregor formula, a well-known integral expression of the m-step transition probability for a nearest-neighbor random walk on the non-negative integers (an infinite path graph), is reformulated in terms of one-mode interacting Fock spaces. A truncated direct sum of onemode interacting Fock spaces is newly introduced and an integral expression for the m-th moment of the associated operator is derived. This integral expression gives rise to an extension of the Karlin-McGregor formula to the graph of paths connected with a clique.

Słowa kluczowe

EN

Jacobi matrix; Karlin-McGregor formula; Kesten distribution; one-mode interacting Fock space; orthogonal polynomials; tridiagonal matrix

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 451--466
• Opis fizyczny: Bibliogr. 17 poz., rys.

Twórcy

autor

Obata N.

• Graduate School of Information Sciences, Tohoku University, Sendai 980-8579 Japan

Bibliografia

• [1] L. Accardi, A. Ben Ghorbal, and N. Obata, Monotone independence, comb graphs and Bose-Einstein condensation, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), pp. 419-435.
• [2] L. Accardi and M. Bożejko, Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 1 (1998), pp. 663-670.
• [3] L. Accardi, H.-H. Kuo, and A. I. Stan, An interacting Fock space characterization of probability measures, Commun. Stoch. Anal. 3 (2009), pp. 85-99.
• [4] L. Accardi and Y. G. Lu, Wiener noise versus Wigner noise in quantum electrodynamics, in: Quantum Probability and Related Topics, VIII, World Sci. Publ., River Edge, NJ, 1993, pp. 1-18.
• [5] M. Bożejko and N. Demni, Generating functions of Cauchy-Stieltjes type for orthogonal polynomials, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), pp. 91-98.
• [6] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.
• [7] P. A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lect. Notes Math., Vol. 3, Amer. Math. Soc., Providence, RI, 1999.
• [8] H. Dette, B. Reuther, W. J. Studden, and M. Zygmunt, Matrix measures and random walks with a block tridiagonal transition matrix, SIAM J. Matrix Anal. Appl. 29 (2006), pp. 117-142.
• [9] F. A. Grünbaum, The Karlin-McGregor formula for a variant of a discrete version of Walsh’s spider, J. Phys. A 42 (2009), 454010, 10 pp.
• [10] H. L. Hamburger, Hermitian transformations of deficiency-index (1, 1), Jacobi matrices and undetermined moment problems, Amer. J. Math. 66 (1944), pp. 489-522.
• [11] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, Math. Surveys Monogr., Vol. 77, Amer. Math. Soc., Providence, RI, 2000.
• [12] A. Hora and N. Obata, Quantum Probability and Spectral Analysis of Graphs, Springer, 2007.
• [13] S. Karlin and J. McGregor, Random walks, Illinois J. Math. 3 (1959), pp. 66-81.
• [14] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), pp. 336-354.
• [15] Y. Kovchegov, Orthogonality and probability: beyond nearest neighbor transitions, Electron. Comm. Probab. 14 (2009), pp. 90-103.
• [16] N. Obata, One-mode interacting Fock spaces and random walks on graphs, Stochastics 84 (2012), pp. 383-392.
• [17] D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monogr. Ser., Amer. Math. Soc., 1992.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).