Stochastic LU factorizations, Darboux transformations and urn models

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

Download Full-text

Some explicit results for correlated random walks

Journal of Applied Probability ◽

10.2307/3214380 ◽

1989 ◽

Vol 26 (4) ◽

pp. 757-766 ◽

Cited By ~ 8

Author(s):

Ram Lal ◽

U. Narayan Bhat

Keyword(s):

Markov Chain ◽

State Space ◽

Equilibrium Distribution ◽

Transition Probabilities ◽

Transition Probability ◽

Transition Probability Matrix ◽

Correlated Random Walks ◽

Finite State ◽

Direction Of Movement ◽

Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

Download Full-text

On the Convergence Rate of Bonus-Malus Systems

Astin Bulletin ◽

10.2143/ast.22.2.2005116 ◽

1992 ◽

Vol 22 (2) ◽

pp. 217-223 ◽

Cited By ~ 6

Author(s):

Heikki Bonsdorff

Keyword(s):

Markov Chain ◽

Probability Distribution ◽

Convergence Rate ◽

Transition Probabilities ◽

Transition Probability ◽

Transition Probability Matrix ◽

Limit Probability ◽

Step Transition

AbstractUnder certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

Download Full-text

Some explicit results for correlated random walks

Journal of Applied Probability ◽

10.1017/s0021900200027625 ◽

1989 ◽

Vol 26 (04) ◽

pp. 757-766 ◽

Cited By ~ 3

Author(s):

Ram Lal ◽

U. Narayan Bhat

Keyword(s):

Markov Chain ◽

State Space ◽

Equilibrium Distribution ◽

Transition Probabilities ◽

Transition Probability ◽

Transition Probability Matrix ◽

Correlated Random Walks ◽

Finite State ◽

Direction Of Movement ◽

Step Transition

Download Full-text

Correlated random walks with stay

Journal of Applied Mathematics and Simulation ◽

10.1155/s1048953388000152 ◽

1988 ◽

Vol 1 (3) ◽

pp. 197-222

Author(s):

Ram Lal ◽

U. Narayan Bhat

Keyword(s):

Markov Chain ◽

Random Walk ◽

Random Walks ◽

Equilibrium Solution ◽

Transition Probability ◽

Transition Probability Matrix ◽

Finite State ◽

Direction Of Movement ◽

One Step ◽

Bivariate Markov Chain

A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability distribution. In a correlated random walk (CRW) the movement follows a Markov chain and induces correlation in the state of the walk at various epochs. Then, the walk can be modelled as a bivariate Markov chain with the location of the particle and the direction of movement as the two variables. In such random walks, normally, the particle is not allowed to stay at one location from one step to the next. In this paper we derive explicit results for the following characteristics of the CRW when it is allowed to stay at the same location, directly from its transition probability matrix: (i) equilibrium solution and the fast passage probabilities for the CRW restricted on one side, and (ii) equilibrium solution and first passage characteristics for the CRW restricted on bath sides (i.e., with finite state space).

Download Full-text

Random walks on free products, quotients and amalgams

Nagoya Mathematical Journal ◽

10.1017/s0027763000000507 ◽

1986 ◽

Vol 102 ◽

pp. 163-180 ◽

Cited By ~ 21

Author(s):

Donald I. Cartwright ◽

P. M. Soardi

Keyword(s):

Random Walk ◽

Probability Measure ◽

Asymptotic Behaviour ◽

Random Walks ◽

Green's Function ◽

Discrete Group ◽

Transition Probabilities ◽

Free Products ◽

Step Transition ◽

Independent And Identically Distributed

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

Download Full-text

Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Entropy ◽

10.3390/e23060729 ◽

2021 ◽

Vol 23 (6) ◽

pp. 729

Author(s):

Miquel Montero

Keyword(s):

Random Walks ◽

Stereographic Projection ◽

Transition Probabilities ◽

Simple Random Walk ◽

General Introduction ◽

Elliptic Case ◽

One Step ◽

Underlying Geometry ◽

Step Transition ◽

Wide Family

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

Download Full-text

Further enhancement on H∞ sliding mode control design for singular Markovian jump systems with partly known transition probabilities

Journal of Vibration and Control ◽

10.1177/1077546321989202 ◽

2021 ◽

pp. 107754632198920

Author(s):

Zeinab Fallah ◽

Mahdi Baradarannia ◽

Hamed Kharrati ◽

Farzad Hashemzadeh

Keyword(s):

Transition Probabilities ◽

Sliding Mode ◽

Transition Probability ◽

Transition Probability Matrix ◽

Linear Matrix ◽

Sliding Mode Controller ◽

Sliding Surface ◽

Markovian Jump ◽

Markovian Jump System ◽

Singular Markovian Jump Systems

This study considers the designing of the H ∞ sliding mode controller for a singular Markovian jump system described by discrete-time state-space realization. The system under investigation is subject to both matched and mismatched external disturbances, and the transition probability matrix of the underlying Markov chain is considered to be partly available. A new sufficient condition is developed in terms of linear matrix inequalities to determine the mode-dependent parameter of the proposed quasi-sliding surface such that the stochastic admissibility with a prescribed H ∞ performance of the sliding mode dynamics is guaranteed. Furthermore, the sliding mode controller is designed to assure that the state trajectories of the system will be driven onto the quasi-sliding surface and remain in there afterward. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design algorithms.

Download Full-text

Transition probability matrices for correlated random walks

Journal of Applied Probability ◽

10.1017/s0021900200046994 ◽

1980 ◽

Vol 17 (01) ◽

pp. 253-258 ◽

Cited By ~ 3

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.

Download Full-text

Stochastic Stability for Time-Delay Markovian Jump Systems with Sector-Bounded Nonlinearities and More General Transition Probabilities

Mathematical Problems in Engineering ◽

10.1155/2013/208263 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Dan Ye ◽

Quan-Yong Fan ◽

Xin-Gang Zhao ◽

Guang-Hong Yang

Keyword(s):

Time Delay ◽

Stochastic Stability ◽

Transition Probabilities ◽

Transition Probability ◽

Sufficient Conditions ◽

Transition Probability Matrix ◽

Markovian Jump Systems ◽

Markovian Jump ◽

Delay Dependent ◽

Jump Systems

This paper is concerned with delay-dependent stochastic stability for time-delay Markovian jump systems (MJSs) with sector-bounded nonlinearities and more general transition probabilities. Different from the previous results where the transition probability matrix is completely known, a more general transition probability matrix is considered which includes completely known elements, boundary known elements, and completely unknown ones. In order to get less conservative criterion, the state and transition probability information is used as much as possible to construct the Lyapunov-Krasovskii functional and deal with stability analysis. The delay-dependent sufficient conditions are derived in terms of linear matrix inequalities to guarantee the stability of systems. Finally, numerical examples are exploited to demonstrate the effectiveness of the proposed method.

Download Full-text

Exploring activity-driven network with biased walks

International Journal of Modern Physics C ◽

10.1142/s012918311750111x ◽

2017 ◽

Vol 28 (09) ◽

pp. 1750111

Author(s):

Yan Wang ◽

Ding Juan Wu ◽

Fang Lv ◽

Meng Long Su

Keyword(s):

Random Walk ◽

Diffusion Process ◽

Random Walks ◽

Stationary Distribution ◽

Significant Role ◽

Search Strategy ◽

Transition Probability ◽

Edge Weighting ◽

Coverage Function ◽

Analytical Expressions

We investigate the concurrent dynamics of biased random walks and the activity-driven network, where the preferential transition probability is in terms of the edge-weighting parameter. We also obtain the analytical expressions for stationary distribution and the coverage function in directed and undirected networks, all of which depend on the weight parameter. Appropriately adjusting this parameter, more effective search strategy can be obtained when compared with the unbiased random walk, whether in directed or undirected networks. Since network weights play a significant role in the diffusion process.

Download Full-text