The relation of Hermite polynomials with the integration theory of Brownian motion was known to N. Weiner during the 1930s followed afterwards by K. Itô. During the 1950s many workers notably contributed to the advancement of the knowledge in the field of establishing ‘important connection between the transition probability functions of diffusion processes, continuous-time birth-death processes and discrete-time birth-death chains by means of a spectral representation’. Prominent were W. Feller [Proc. Berkeley Sympos. Math. Statist. Probability, California July 31–August 12, 1950, 227–246 (1951; Zbl 0045.09302); J. Math. Pures Appl. (9) 38, 301–345 (1959; Zbl 0090.10902)], S. Karlin and J. McGregor [Trans. Am. Math. Soc. 85, 489–546 (1957; Zbl 0091.13801); Ill. J. Math. 3, 66–81 (1959; Zbl 0104.11804)], among others. The present monograph is devoted to an excellent discussion of the most important results developed during the past six decades or more which are ‘related to the spectral representation of the most important one-dimensional Markov processes, namely discrete-time birth-death chains (random walks), birth-death processes and diffusion processes’. The book, on the one hand can be found much resourceful by statisticians who are well familiar with stochastic processes but may be not very familiar with the theory of special functions and orthogonal polynomials, while on the other hand it can also be most fruitfully utilized by researchers working in the field of special functions and orthogonal polynomials to study the deep rooted interconnections between the Markov processes and orthogonal polynomials.
The research monograph is divided into four chapters. Since the statisticians are well versed with the probability theory and the stochastic processes, the author introduces orthogonal polynomials in the very first chapter of the book, which also contains an enriching discussion on the Stieltjes transform and its properties, which is later on widely used to study ‘the spectral analysis of discrete-time birth-death chains and birth-death processes’ Besides discussing the Favard’s theorem, this chapter also discusses the classical families of orthogonal polynomials of a continuous variable (Hermite, Laguerre and Jacobi – with the special cases of the Jacobi polynomials – Gegenbauer (ultraspherical), Legendre and Chebychev) as well as a discrete variable (Charlier, Meixner, Krawtchouk and Hahn) which are both characterized by the fact that both these families are ‘are eigenfunctions of a second-order differential operator (in the continuous variable) or a second-order difference operator (in the discrete variable) of the Sturm-Liouville type’. The Askey scheme of orthogonal polynomials is also discussed in the last section of this chapter with brief discussions on the dual Hahn, Racah, Wilson, continuous Hahn, continuous dual Hahn and Meixner-Pollaczek polynomials.
The second chapter discusses the spectral analysis (probabilistic and asymptotic aspects) of discrete-time birth-death chains on \(\mathbb{N}_0\), which are characterized by a tridiagonal one-step transition probability matrix. The author prefers to use the term ‘discrete-time birth-death chains’ for these processes, instead of the other used term ‘random walks’ for these processes because random walks usually are ‘discrete-time birth-death chains with constant transition probabilities’. After giving the basics of discrete-time Markov chains in the first section of the chapter, the author discusses the famous ‘Karlin-McGregor integral representation formula of the \(n\)-step transition probabilities \(P_{ij}^{(n)}\) in terms of orthogonal polynomials with respect to a probability measure \(\psi\) with support inside the interval \([-1,1]\)’. In the third section of this chapter the author discusses the theory of some polynomial families like the dual polynomials, the \(k\)th associated polynomials and the Derman-Vere-Jones transformation which are used in the next section of the chapter where some well known examples of the discrete-time birth-death chains – like the Gambler’s ruin, Ehrenfest urn model (which is related to the Orstein-Uhlenbeck diffusion process in Chapter 4) and the Bernoulli-Laplace urn model – all of which are examples with finite state space \(\mathcal{S}=\{0,1,\ldots, N\}\). Some examples like the Gambler’s ruin on \(\mathbb{N}_0\) with one absorbing barrier, etc. using the Stieltjes transform are also discussed. An example of a birth-death chain generated by the Jacobi polynomials is also dealt with followed by a section on the study of the probabilistic and asymptotic aspects of a discrete-time birth-death chain by applying the Karlin-McGregor integral representation formula including the recurrence and absorption and the strong ratio limit property. This chapter also discusses the concept of limiting conditional distribution and the spectral representation of discrete-time birth-death chains on \(\mathbb{Z}\).
The third chapter deals with the spectral analysis of birth-death processes on \(\mathbb{N}_0\) which are characterized by an infinitesimal operator – a tridiagonal matrix with spectrum inside the interval \((-\infty,0]\). The Karlin-McGregor integral representation formula of the transition probability functions \(P_{ij}(t)\) in this case is deduced ‘in terms of orthogonal polynomials with respect to a probability measure \(\psi\) with support inside the interval \([0,\infty)\)’. The results in this case are similar to their counterparts in Chapter 2 yet the techniques and methods employed here are different in the sense that in the Section 3.4 it is proven rigorously that ‘the Karlin-McGregor representation formula is in fact a transition probability function of a birth-death process’. The Section 3.3 in which the properties of the dual polynomials and the \(k\)th associated polynomials are studied for the birth-death processes almost parallels the Section 2.3. Birth-death processes with killing are discussed in the Section 3.5. Examples with finite space like the \(M/M/1/N\) queue, the Ehrenfest and Bernoulli-Laplace urn models and a genetic model of Moran and examples with state space \(\mathbb{N}_0\) like the \(M/M/1\) queue, the \(M/M/\infty\) queue, the \(M/M/k\) queue, the linear birth-death process, etc. are thoroughly discussed in detail in Section 3.6. The Karlin-McGregor integral representation formula is invoked in Section 3.7 to explore the probabilistic and asymptotic aspects of any birth-death process which includes the recurrence and absorption, the strong ratio limit property and the limiting conditional distribution (as in Chapter 2), the decay parameter and the quasi-stationary distributions. Finally, in Section 3.8 the author discusses the spectral representation of birth-death processes on \(\mathbb{Z}\) (the bilateral birth-death processes).
In the concluding chapter an exhaustive discussion of the spectral analysis of one-dimensional diffusion processes for which ‘the state space is a continuous interval contained in \(\mathbb{R}\)’, is given. For these processes an infinitesimal operator is a ‘second-order differential operator with a drift and a diffusion coefficient’. A basic overview of the diffusion processes is given in Section 4.1 followed by the deduction of ‘spectral representation of the transition probability density \(p(t;x,y)\) in terms of the orthogonal eigenfunctions of the corresponding infinitesimal operator of the diffusion process’ in Section 4.2. The classification of boundary points with reflecting and absorbing boundaries and the Feller’s classification of boundaries is discussed in Section 4.3, which gives the boundary conditions for solving certain regular Sturm-Liouville problems associated with the diffusion processes. The spectral decomposition of a diffusion process with killing is investigated in Section 4.4. A number of examples of diffusion processes are studied in detail in Section 4.5 which include the Brownian motions with drift and scaling, examples related to classical orthogonal polynomials like the Orstein-Uhlenbeck process, a population growth model, the Jacobi diffusion process, Brownian motion on the sphere and Gegenbauer polynomials, a population genetics diffusion process with killing, radial Brownian motion in \(N\)-dimensions, radial Orstein-Uhlenbeck process in \(N\) dimensions and finally examples with absolutely continuous and discrete spectrum. At the end section of this chapter an interesting discussion of quasi-stationary distributions is provided.
The book serves as an excellent research monograph in this field and is strongly recommended by the reviewer to the researchers working in this field - both statisticians and mathematicians. The reviewer would also like to mention here for the sake of interested researchers in this and allied fields of study that with the properties of the Upadhyaya transform (see, [L. M. Upadhyaya, “Introducing the Upadhyaya integral transform” Bull. Pure Appl. Sci. E, Math. Stat. 38E, No. 1, 471–510 (2019)] and [L. M. Upadhyaya et al., “An update on the Upadhyaya transform”, Bull. Pure Appl. Sci. E, Math. Stat. 40E, No. 1, 26–44 (2021)] which is, till date, the most powerful and the most versatile generalization of the various variants of the Laplace type of integral transforms introduced in the mathematics research literature, the further generalization of the properties of the resolvent function mentioned in the equation (3.12) for continuous time Markov chains, the Karlin-McGregor representation formula for birth-death processes of Theorem 3.9 and the results of Proposition 3.19 of this book can be an interesting direction of future research investigations in this field.