论文信息 - Random walk polynomials and random walk measures

Random walk polynomials and random walk measures

Random walk polynomials and random walk measures play a prominent role in the analysis of a class of Markov chains called random walks. Without any reference to random walks, however, a random walk polynomial sequence can be defined (and will be defined in this paper) as a polynomial sequence{Pn(x)} which is orthogonal with respect to a measure on [-1, 1] and which is such that the parameters (alfa)n in the recurrence relations Pn=1(x)=(x(alfa)n)Pn(x)-snPn-1(x) are nonnegative. Any measure with respect to which a random walk polynomial sequence is orthogonal is a random walk measure. We collect some properties of random walk measures and polynomials, and use these findings to obtain a limit theorem for random walk measures which is of interest in the study of random walks. We conclude with a conjecture on random walk measures involving

Erik A. van Doorn | Pauline Schrijner

[1] J. Shohat,et al. The problem of moments , 1943 .

[2] Thomas Whitehurst. An application of orthogonal polynomials to random walks. , 1982 .

[3] Strong ratio limit property and R-recurrence of reversible Markov chains , 1974 .

[4] E. V. Doorn,et al. On Oscillation Properties and the Interval of Orthogonality of Orthogonal Polynomials , 1982 .

[5] T. A. Bromwich. An Introduction To The Theory Of Infinite Series , 1908 .

[6] T. Chihara,et al. An Introduction to Orthogonal Polynomials , 1979 .

[7] Paul Neval,et al. Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study , 1986 .

[8] Erik A. van Doorn,et al. Representations and bounds for zeros of Orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices , 1987 .