On the generating functions of a random walk on the non-negative integers

1996 ◽  
Vol 33 (4) ◽  
pp. 1033-1052 ◽  
Author(s):  
Holger Dette
Keyword(s):  
Random Walk ◽  
Orthogonal Polynomials ◽  
State Space ◽  
Generating Functions ◽  
Spectral Measure ◽  
Stieltjes Transform ◽  
Explicit Representations ◽  
Step Transition

In the random walk whose state space is a subset of the non-negative integers explicit representations for the generating functions of the n-step transition and the first return probabilities are obtained. These representations involve the Stieltjes transform of the spectral measure of the process and the corresponding orthogonal polynomials. Several examples are given in order to illustrate the application of the results.

On the generating functions of a random walk on the non-negative integers

1996 ◽  
Vol 33 (04) ◽  
pp. 1033-1052
Author(s):  
Holger Dette
Keyword(s):  
Random Walk ◽  
Orthogonal Polynomials ◽  
State Space ◽  
Generating Functions ◽  
Spectral Measure ◽  
Stieltjes Transform ◽  
Explicit Representations ◽  
Step Transition

In the random walk whose state space is a subset of the non-negative integers explicit representations for the generating functions of then-step transition and the first return probabilities are obtained. These representations involve the Stieltjes transform of the spectral measure of the process and the corresponding orthogonal polynomials. Several examples are given in order to illustrate the application of the results.

Spectral measure of the Thue–Morse sequence and the dynamical system and random walk related to it

2015 ◽  
Vol 36 (4) ◽  
pp. 1247-1259
Author(s):  
LI PENG ◽  
TETURO KAMAE
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Spectral Measure ◽  
Correlation Dimension ◽  
Transition Probability ◽  
Stat Phys ◽  
Kolmogorov Type ◽  
Left And Right ◽  
Step Transition ◽  
Image Position

Let $1,-1,-1,1,-1,1,1,-1,-1,1,1,\ldots$ be the $\{-1,1\}$-valued Thue–Morse sequence. Its correlation dimension is $D_{2}$, satisfying $$\begin{eqnarray}\mathop{\sum }_{k=0}^{K-1}|{\it\gamma}(k)|^{2}\asymp K^{1-D_{2}}\end{eqnarray}$$ in the sense that the ratio between the left- and right-hand sides is bounded away from 0 and $\infty$ as $K\rightarrow \infty$, where ${\it\gamma}$ is the correlation function; its value is known [Zaks, Pikovsky and Kurths. On the correlation dimension of the spectral measure for the Thue–Morse sequence. J. Stat. Phys.88(5/6) (1997), 1387–1392] to be $$\begin{eqnarray}D_{2}=1-\log \frac{1+\sqrt{17}}{4}\bigg/\log 2=0.64298\ldots .\end{eqnarray}$$ Under its spectral measure ${\it\mu}$ on $[0,1)$, consider the transformation $T$ with $Tx=2x$ ($\text{mod}~1$). It is shown to be of Kolmogorov type having entropy at least $D_{2}\log 2$. Moreover, a random walk is defined by $T^{-1}$ which has the transition probability $$\begin{eqnarray}P_{1}((1/2)x+(1/2)j\mid x)=(1/2)(1-\cos ({\it\pi}(x+j)))\quad (j=0,1).\end{eqnarray}$$ It is proved that this random walk is mixing and ${\it\mu}$ is the unique stationary measure. Moreover, $$\begin{eqnarray}\lim _{N\rightarrow \infty }\int P_{N}((x-{\it\varepsilon},x+{\it\varepsilon})|x)\,d{\it\mu}(x)\asymp {\it\varepsilon}^{D_{2}}\quad (\text{as}~{\it\varepsilon}\rightarrow 0),\end{eqnarray}$$ where $P_{N}(\cdot \mid \cdot )$ is the $N$-step transition probability.

SYMMETRIC FUNCTIONS OF BINARY PRODUCTS OF TRIBONACCI LUCAS NUMBERS AND ORTHOGONAL POLYNOMIALS

2021 ◽  
Vol 21 (2) ◽  
pp. 461-478
Author(s):  
HIND MERZOUK ◽  
ALI BOUSSAYOUD ◽  
MOURAD CHELGHAM
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Symmetric Functions ◽  
Lucas Numbers

In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

On Sieved Orthogonal Polynomials II: Random Walk Polynomials

1986 ◽  
Vol 38 (2) ◽  
pp. 397-415 ◽  
Author(s):  
Jairo Charris ◽  
Mourad E. H. Ismail
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Orthogonal Polynomials ◽  
Transition Probabilities ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Markov Process ◽  
Image Position

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

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