Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 14 (2009) > Paper 9 open journal systems 


Orthogonality and probability: beyond nearest neighbor transitions

Yevgeniy V Kovchegov, Oregon State University


Abstract
In this article, we will explore why Karlin-McGregor method of using orthogonal polynomials in the study of Markov processes was so successful for one dimensional nearest neighbor processes, but failed beyond nearest neighbor transitions. We will proceed by suggesting and testing possible fixtures.


Full text: PDF

Pages: 90-103

Published on: February 16, 2009
Bibliography
  1. N.I.Aheizer and M.G.Krein. Some questions in the theory of moments. Translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2 American Mathematical Society, Providence, R.I. (1962) v+265 pp. MR0167806(29 #5073)
  2. P.Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Amer. Math. Soc., Providance, RI, (2000). MR1677884
  3. P.Deift. Riemann-Hilbert Methods in the Theory of Orthogonal Polynomials Spectral Theory and Mathematical Physics 76 (2006), 715-740, Amer. Math. Soc., Providance, RI MR2307753
  4. H.Dym and H.P.McKean. Gaussian processes, function theory, and the inverse spectral problem Probability and Mathematical Statistics 31 (1976), 435-451. Academic, New York - London MR0448523
  5. M.L.Gorbachuk and V.I.Gorbachuk. M.G.Krein's Lectures on Entire Operators Birkhaüser Verlag (1997) Math. Review number not available.
  6. F.A. Grünbaum. Random walks and orthogonal polynomials: some chal lenges Probability Geometry and Integrable Systems - MSRI Publications 55 (2007), 241-260 MR2407600
  7. S.Karlin. Total Positivity Stanford University Press, Stanford, CA (1968) MR0230102
  8. S.Karlin and J.L.McGregor. The differential equations of birth and death processes, and the Stieltjes moment problem Transactions of AMS, 85, (1957), 489-546 MR0091566
  9. S.Karlin and J.L.McGregor. The classification of birth and death processes Transactions of AMS, 86, (1957), 366-400 MR0094854
  10. S.Karlin and J.L.McGregor. Random Walks Illinois Journal of Math., 3, No.1, (1959), 417-431. MR0100927
  11. S.Karlin and J.L.McGregor. Occupation time laws for birth and death processes Proc. 4th Berkeley Symp. Math. Statist. Prob., 2, (1962), 249-272 MR0137180
  12. S.Karlin and J.L.McGregor. Linear Growth Models with Many Types and Multidimen- sional Hahn Polynomials R.A. Askey, Editor, Theory and Applications of Special Functions, Academic Press, New York, (1975), 261-288 MR0406574
  13. Y.Kovchegov, N.Meredith and E.Nir Occupation times via Bessel functions preprint Math. Review number not available.
  14. A.B.J.Kuijlaarsr. Riemann-Hilbert Analysis for Orthogonal Polynomials Orthogonal Poly- nomials and Special Functions (Springer-Verlag), 1817, (2003) MR2022855
  15. W.Schoutens. Stochastic Processes and Orthogonal Polynomials. Lecture notes in statis- tics (Springer-Verlag), 146, (2000) MR1761401
  16. G.Szegö. Orthogonal Polynomials. Fourth edition. AMS Colloquium Publications, 23, (1975) MR0372517
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X