Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata

Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0, such that the increments $\varepsilon_{j}:=\mathbf{X}_{j}-\mathbf{X}_{j-1}$ take values $\pm1$ and satisfy $P(\varepsilon_{j+1}=1|\varepsilon_{j}=1, |\mathbf{X}_{j}|=k)=P(\varepsilon_{j+1}=-1|\varepsilon_{j}=-1,|\mathbf{X}_{j}|=k)=a_k$, all $j\ge 1$, where $a_k=a$ if $0\le k\le f-1$, and $a_k=b$ if $f\le k<N$. Here $0<a, b <1$ denote persistence parameters and $f ,N\in \mathbb{N}$ with $f<N$. The process starts at $\mathbf{X}_0=m\in (-N,N)$ and terminates when $|\mathbf{X}_j|=N$. Denote by ${\cal R}'_N$, ${\cal U}'_N$, and ${\cal L}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left ({\cal L}'_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}'_N-\frac{1}{(1-a)(1-b)}{\cal U}'_N\right )/N$ and let $f\sim\eta N$ for some $0<\eta <1$. We show $\lim_{N\to\infty} E\{e^{itX_N}\}=\hat{\varphi}(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been streamlined, with references added, including reference to a companion document with details of calculations via Mathematica. The 3rd version has 2 new figures and improved presentatio

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [−1, 1] with weight function: w(x)=(1−x)α(1+x)βh(x),α,β>−1 and h(x) a real, analytic and strictly positive function on [−1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n2) based on the recurrence relation

Orthogonal polynomial interpretation of q-Toda and q-Volterra equations

The correspondences between dynamics of q-Toda and q-Volterra equations for the coefficients of the Jacobi operator and its resolvent function are established. The orthogonal polynomials associated with these Jacobi operators satisfy an Appell condition, with respect to the q-difference operator Dq . Lax type theorems for the point spectrum of the Jacobi operators associated with these equations are obtained. Examples related with the big q-Legendre, discrete q-Hermite I, and little q-Laguerre orthogonal polynomials and q-Toda and q-Volterra equations are given