S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes. Academic Press, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....process is a Markovian process in continuous time which is completely described by the transition rates, P ; 1 , P ; and P ; Gamma1 . The replicationdeletion process with the approximations adopted above, and with a finite population of constant size N , is a birth and death process [25, 35, 36] that converges to a stationary distribution, defined by: dP =dt = 0 8 . The time axis can be transformed such that the probability for a replication deletion event becomes independent of the population average. This reactor time t is defined by d t = dt (oe Gamma 1) N = dt ( ....

S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981.

....functions. Recall that if a queue service distribution (specifying the number of customers served in a time step) has expectation and standard deviation # (both of which we assume to be finite) then the expected waiting time under Poisson arrivals with rate # is #(1 # 2( see [80] or [94] for a derivation. To rephrase this formula in our usual notation, we view the parameter as the edge capacity u and the Poisson rate # as the amount of tra#c assigned to an edge; we are then interested in latency functions # of the following form: #(x) x(1 # 2u(u . As in the ....

S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.

.... h) X(t) X(t) x] 18) which are called the drift parameter and the infinitesimal variance, respectively. For our purposes it is sufficient to consider the time homogenous processes for which the infinitesimal parameters depend only on the state x and not on t. Following the procedure in [17], we now set the state 0 and the predefined overflow limit c as absorbing barriers. The problem is to find the probability of absorption into c when starting from the position X(0) x,0 x c. Denote this probability by u(x) and select a small time duration h,sosmall that the absorption ....

....zero, we get a differential equation for the desired probability. 0 = x) du (x) 0 x c, u(0) 0, u(c) 1. 20) Assume that s (x) 0 and define the scale function of the process as S(x) exp 2(h) h) dh dt. 21) Then the general solution of (20) can be written [17] as S(x) S(c) 0 1. 22) The result given here can be easily applied to provide results for all time homogenous diffusion processes. For the simplest model, we follow the example of Harrison and Patel [18] and shed light on a GI GI 1 queuing model with independent inter arrival time ....

Samuel Karlin and Howard M. Taylor, A Second Course in Stochastic Processes, Academic Press, San Diego, 1981.

....of allowable latency functions. Recall that if a queue service distribution (specifying the number of customers served in a time step) has finite expectation and finite standard deviation #, then the expected waiting time with Poisson arrivals with rate # is #(1 # 2( see [8] or [9] for a derivation. To rephrase this formula in our usual notation, we view the parameter as the edge capacity u and the Poisson rate # as the amount of tra#c assigned to an edge; we are then interested in latency functions # of the following form: #(x) x(1 # 2u(u x) As in the ....

S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.

....noise in the form of a Wiener or Gaussian process. The addition of random jumps and later Poisson processes became possible through the original work of theorists such as Florentin [ It6 [13] Kushner [16] and Gihman and Skorohod [8] and continued by people such as Karlin and Taylor [14], Ryan and Hanson [19] Marlton [18] and Westman and Hanson[20, 22] One general approach is to use stochastic dynamic programming, but this requires a large amount of memory. Large random jumps can be especially troublesome, even when finite elements are used, in part due to the Curse of ....

S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.

....v t v x v y is supposed to follow the stochastic infinitesimal equation (M1) dv t av t dt for positive constants a and b and a two dimensional Brownian motion W t . An introduction to SDEs can be found in [43] and an elegant treatment of one dimensional diffusions is given in [31]. The coefficient av t is called the infinitesimal drift and b the infinitesimal variance. Heuristically, a gives the rate at which the velocity regresses to zero and b gives the magnitude of the random innovations which tend to push the velocity away from zero. The root mean square speed ....

....of the bacterium up an increasing gradient of a ligand, and f : is a convex function, then the expected value E f x t is monotone increasing with time. This would also be true if x t were an unbiased random walk, or a martingale. Comparing the Ito and Stratonovich solutions [31] to the SDEs introduced in Section 3 gives a new insight into the controversy. Kinesis may be modeled in a simple but instructive way by considering a stochastic process x t , taking values in 0 , defined by the infinitesimal equation dx t sx t dW t This is a particular case of (M4) ....

S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981.

....q distribution) on the set of vertices does not seem to be of simple nature and no substantial properties are known. Here we consider an extension of the second approach which is easily implementable by distributed algorithms. The idea is based on the assumptions linked to Poisson processes, see [4]. Each vertex which has become a leaf, has lifetime which is a real valued random variable exponentially distributed. The leaf is removed once its lifetime has expired (if the tree is not reduced at that time to a unique vertex) The process continues until the tree is reduced to a vertex; the ....

....are functions of the initial one and the communicated values ineherited from the removed adjacent leaves. 4 Simple Properties of the General Model According to the previous section, the general model (Algorithm 2) may be regarded as a continueoustime pure death stochastic process on a tree, see [4]. Once a vertex v has become a leaf, its death happens according to a Poisson process with a given parameter (v) which is supposed to have been determined by some speci c rules detailed in the sequel. Each leaf v of the residual tree behaves independently and, under the assumption that it is ....

[Article contains additional citation context not shown here]

S. Karlin and H. M. Taylor. A second course in stochastic processes. Academic Press, 1981.

....di usion term and bounded drift and with re ected barrier hits any point in a compact set in nite time with probability one, we have: Corollary 13 When 6= 0, under the optimal play U (x) Pf 1g = 1. 2.3. 2 Ruin probability for = 0 Adopting the terminology in Section 6, Chapter 15 of [9], we start by checking S(x) for x x , where S(x) Z x 0 s(y)dy = Z x 0 exp 2 Z y U (y) U (y) 2 ( U (y) 2 dy: 32) If S(x) 1 for some x x , then the ruin probability is zero. Otherwise, it means that zero is an attracting point, and we need to ....

.... ( 2 s( d = lim l 0 C Z x l 2 6 6 4 Z l exp 2 2 exp 2 2 1 2 2 2 2r 2 3 7 7 5 exp 2 2 exp 2 2 1 2 2 2 2 2r 2 d = A lim l 0 ln exp 2 l 2 1 = 1 Here A is bounded. Hence by Lemma 6. 2 ([9], page 230) we have Corollary 14 When = 0, Pf 1g = 0 under the optimal strategy U (x) In particular, zero is an attractive and unattainable point when 2 2r 2 . It is worth pointing out that in [10] and [12] the statement that a company goes bankrupt in nite time with ....

S. Karlin and H. Taylor (1981): A Second Course in Stochastic Processes. Academic Press.

....k=1 g n;k b k;0 : Remark 1 i) As a generalization, we can define a i;j and b j;i for all i and j. Then for any fixed i, a i;j and b j;i will play the same role as a 0;i and b i;0 . ii) a = u Q 1. When the Markov chain is recurrent, this result can also be found in Section 3 of Chapter 11 of [8] or Theorem 3.3 of [1] If the Markov chain is transient, then P 1 k=0 P k 1 (for example, see the last equation on page 107 in Ross [16] This means that the expected number of visits to state n starting from 0 is finite. Therefore, the expected number of visits to state n before hitting ....

....Markov chain, ergodic or nonergodic, the convergence of (n) b to b in Theorem 4 is also in the sense of l 1 . For any ergodic Markov chain, the convergence of (n) a to a in Theorem 4 is also in the sense of l 1 . When the Markov chain P is ergodic, then 0 a 0;j = j according to Theorem 3. 3 of [8]. Using a similar argument (also see Remark 1) we can show that i a i;j = j . Remark 2 i) When the Markov chain P is ergodic, then a is proportional to the stationary probability vector ( 1 ; 2 ; and (a 0;0 ; a 0;1 ; a 0;2 ; with a 0;0 = 1 is one of the regular measures of P . ....

Karlin, S.K. and Taylor, H.M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.

....P 1 n=1 R n Gamma1;n 1. Proof: This follows from Corollary 2.7 and Theorem 2.1. Remark 2.9 i) When P is ergodic, 0 (A 0;0 ; A 0;1 ; A 0;2 ; is the stationary probability vector of P , unique up to multiplication by a constant. ii) A i;j was introduced by Karlin and Taylor, page 35 of [7], for the case where P has scalar entries. They used it to study ratio theorems and for the interpretation of generalized stationary probabilities. Theorem 2.10 Matrices B n;0 and G n;i as defined in Section 1 satisfy B n;0 = 8 : G 1;0 ; if n = 1; G n;0 P n Gamma1 i=1 G n;i B i;0 ; if n ....

Karlin, S. and Taylor, H.M. (1981) A Second Course in Stochastic Processes, Academic Press, New York.

....l customers arrivetothe system. Let ae (k) denote the traffic intensityofthekth source, whichisgiven by ae (k) k) 1 X l=1 lA (k) l e M (k) 1) Recall that the underlying Markov chain of the kth source is periodic with period R (k) Let D (k) r denote the rth moving class [14] of the kth source, i.e. if i 2 D (k) r (i =1#: #M (k) #r =1#: #R (k) then P (k) fS (k) n 1 = jjS (k) n = ig =0for all j 62 D (k) r Phi k 1 , where Phi k (k =1#: #K) is defined as r Phi k l = r l ; 1) mod R (k) 1: Note that D (k) r (r = 1#: #R (k) gives an exact ....

S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, London, 1981.

....if and if only P (n) is recurrent for all n 1. ii) a ij 1. To see this, without loss of generality, consider a = a 01 ; a 0;2 ; When the Markov chain is ergodic the result is obvious. When the Markov chain is recurrent, this result can also be found in Section 3 of Chapter 11 of [5]. When the Markov chain is transient, then P 1 k=0 P k 1. This means that the expected number of visits to state n starting from 0 is finite. Therefore, the expected number of visits to state n before hitting state 0 should also be finite. Theorem 1 For every infinite irreducible ....

....upper triangular, a direct calculation shows that A Gamma I has a unique inverse. Let E = A Gamma I) Gamma1 ED . Then I Gamma P = A Gamma I) B Gamma S E) where E = 0 if and only if P is recurrent. iii) The generalized stationary probability distribution or left regular vector (see [5]) can be computed numerically with this decomposition (following [4] for the null recurrent case. Acknowledgements This work has been supported by grants from the Natural Sciences and Engineering Research Council of Canada. The authors thank the referee for the useful comments and suggestions. ....

S. Karlin and H. Taylor, A Second Course in Stochastic Processes, Academic Press, San Diego, 1981.

....of a corresponding potential function is that y H x = x H y (2:3) 25, 26] In the case that the region is simply connected, this condition is also sufficient. 2. 2 Stochastic case: A pertinent probabilistic concept for dynamic situations is a stochastic differential equation (SDE) see [3, 16]. Such equations lead to Markov processes and take the form dr(t) r(t) t)dt Sigma(r; t)dB(t) 2 :4 ) 6 with the drift parameter, Sigma the variance or diffusion parameter and B bivariate Brownian motion. Here r; B are vectors while Sigma is a matrix. The parameters have the ....

Karlin, S. and Taylor, H.M. (1981). A Second Course in Stochastic Processes. Academic, New York.

....exp ## x 0 2(z) # 2 (z) dz # # 2 (x) 4.2) where m is the speed density, s the scale function. We call a di#usion process positiverecurrent if # # # m(x)dx #, which is equivalent to #T # #, where T is the first exit time of ( #,V thre ] For a positive recurrent process[23], its stationary distribution density is given by #(x) # m(x) The following conclusion is proved in [13] Theorem 1 For a positive recurrent di#usion process X t we have #T # = 2 # V thre Vrest s(u)du # Vrest # m(u)du 2 # V thre Vrest # # V thre y s(u)du # m(y)dy = 2 # V ....

Karlin S. and Taylor H.M. (1982) A Second Course in Stochastic Processes Academic Press, New York.

.... drift B and local variance V , let T a = infft : Z(t) ag and v(z) E(T a jZ(0) z) Then, v is the unique solution to the linear ODE 1 2 V d 2 v dz 2 Bz dv dz 1 = 0; with v(a) 0 (an absorbing barrier at a) and v 0 (b) 0 (a reflecting barrier at b) see Section 15.3 of Karlin and Taylor [1981]) For the logistic model, we have V = 2 and B = The DE was solved by first converting it to two firstorder ODEs in the usual way, and then using the Matlab (Version 6) command bvp4c, which implements a three stage Lobatto IIIa formula to solve two point boundary value problems. Note that the ....

Karlin, S. and Taylor, H. (1981) A Second Course on Stochastic Processes, Acad. Press, London.

....no restriction) is that (x) is hereditary, that is (x) 0 ) y) 0 if y x. 2.1) where y x means that y is a sub configuration of x. 2.1 Markov chains general theory In this section a short review of Markov Chains is given. For a more detailed discussion, we refer to Karlin and Taylor [22, 23]. Consider a sequence of stochastic variables X(1) X(2) 2 Omega Gamma The stochastic process fX(s)g 2 is said to be a Markov chain if 1 Actually, simulation based on Markov chains started with Metropolis and co workers in 1953, but the high interest in such techniques has mainly been ....

S. Karlin and H. M. Taylor. A second course in stochastic processes. Academic Press, 1981.

....processes, fX(t) t 0g: Consider the following hypotheses: H 0 : X(t) is a diusion process solution to the following stochastic dierential equation: dX(t) b(X(t) dt (X(t) dW (t) for some = 0 2 ; where 2 k is compact. 3 HA : The negation of H 0 : It is known (e.g. see Karlin and Taylor (1981) pp. 241) that for a given initial condition, the drift and variance terms (b( and 2 ( respectively) uniquely determine the stationary density associated with the invariant probability measure of the above diusion process, say f(x; In particular, f(x; exp R x l [2b( 2 ( ....

Karlin, S. and H.M. Taylor, (1981), A Second Course in Stochastic Processes, Academic Press, San Diego.

.... Gamma )b) B=A) we see that G 0 (fl) 0 implies that oe S oe 0 S decreases with fl. Xi Appendix 2. Proof of Proposition 7 The Fourier Transform of the state return density function over primitive states can be characterized by the usual Fokker Planc backward equation (c.f. for example Karlin and Taylor 1982), which is the fundamental PDE above (30) along with the initial condition f(z t ; t ; 0) ffi z Gammaz t , where ffi stands for the Dirac function, and the last argument of f( Delta; Delta) denotes time to maturity. Substituting z = log S into and letting j T Gamma t be the time to maturity ....

....by a series of length M, the set of M 2 equations are recursively solved for an , n = 2; Delta Delta Delta ; M , as functions of a 0 , and a 1 . The boundary conditions provide the two remaining equations to solve the ODE. Xi Proof of Lemma 3. We closely follow the analysis in Chapter 15 of Karlin and Taylor (1982). In the two state case the posterior probability follows the diffusion process d = dt oe( dW (52) The stationary density, Psi(x) of the diffusion process (52) must satisfy 1 2 2 y 2 [oe 2 (y) Delta (y) Gamma y [ y) Delta (y) 0; 46 where under the risk neutral ....

Karlin, Samuel and Howard M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1982.

....some new targeted product. Assume that r varies randomly as a g W(t) where W(t) is a Gaussian white noise process with zero mean, a, g being positive constants. Then one can derive the infinitesimal mean (x) and variance s 2 (x) of the stochastic process x t as follows: see, e.g. Karlin and Taylor (1981) (x) x(a ax) 7) s 2 (x) g 2 x 2 On substitution the mean (x) can be expressed as a function of variance s 2 (x) x) as(x) g) as 2 (x) g 2 It follows that ) x ( x ( 2 s 0, if s 2 (x) ag (2a) 2 (8) Otherwise (x) s 2 (x) 0. If output is proportional to ....

Karlin, S. and Taylor, H.M. (1981). A Second Course in Stochastic Processes, New York, Academic Press.

....) with V positive, stationary and ergodic: 1 Assumption 2. 1 For any value of q 2 Q (A1) there exists a unique strong solution (X ;V ) to (3) 4) with state space R (0; and initial value (X 0 ;V 0 ) U X ;U V ) 1 Simple integral conditions ensuring stationarity can be found in Karlin Taylor (1981, Section 15.6) or Karatzas Shreve (1991, Section 5.5) for example. Approximations to the likelihood function 5 (A2) the process V is stationary and ergodic with invariant measure q and V 0 =U V q . It is natural to consider increments of X . Define for i 2 N the random ....

Karlin, S. & Taylor, H. M. (1981), A second course in stochastic processes, Academic Press, New York.

.... Gamma 1)oe 2 2ff E[X m Gamma1 ] 43) for m 2. The spectrum (set of eigenvalues) is = fjff : j 2 N 0 g with corresponding eigenfunctions OE j (x; L (j) j (2ffxoe Gamma2 ) where L (j) j is the jth order Laguerre polynomial with parameter j = 2fffioe Gamma2 Gamma 1 (Karlin and Taylor, 1981). A discretized trajectory (Y t i ) 0in with t i = i Delta is assumed to be given by Y t i = X t i t i : 44) For notational simplicity, we assume that only one parameter is to be estimated, i.e. p = 1. 12 Let f(y) y 2 , J = 1, and let the space P i Gamma1 on which the linear ....

Karlin, S. and Taylor, H. M. (1981), A Second Course in Stochastic Processes, Academic Press.

....= x; dx on I such that X is strictly stationary and ergodic if X 0 ; 3. the drift function b is in L 1 ( for all 2 . Since X is continuous, the state space I is an interval and we write I = l; r) where 1 l r 1. Simple integral conditions ensure stationarity, see Karlin and Taylor (1981, Section 15.6) or Karatzas and Shreve (1991, Section 5.5) for example: De ne the scale density s( by log s(x; 2 R x x0 b(u) 2 (u; du where x 0 2 I is xed but arbitrary. If 1=K 0 ( R r l (s(x; 2 (x; 1 dx 1 and Z x0 l s(x; dx = Z r x0 s(x; dx = 1 ....

Karlin, S. and Taylor, H. M. (1981), A Second Course in Stochastic Processes, Academic Press, New York.

....clear proof is given of the fact that A is of the form: Af(x) x)f 0 (x) 1 2 oe 2 (x)f 00 (x) where (x) is called the drift coefficient and oe 2 (x) 0 the diffusion coefficient. We will highlight the spectral representation for some of diffusion processes in the examples (see also [21]) 2.2 Stein s Method 2.2.1 Normal Approximation and Poisson Approximation In 1972, Stein [28] published a method to prove Normal approximation. It is based on the fact that a random variable Z has a Standard Normal distribution N(0; 1) if and only if for all differentiable functions f such that ....

....(x) x) Gammax and oe 2 (x) 2s(x) 2, then we have AH n (x= p 2) GammanH n (x= p 2) where the operator A is given by Af = f 00 (x) Gamma xf 0 (x) This is the generator of the the Ornstein Uhlenbeck Process. The spectral representation for the transition density is given by [21] p(t; x; y) e Gammay 2 =2 p 2 1 X n=0 e Gammant H n (x= p 2)H n (y= p 2) 1 2 n n : where H n (x) is the Hermite polynomial of degree n [22] The Hermite polynomials H n (x= p 2) are orthogonal with respect to the Standard Normal distribution we started with. 2. The Gamma ....

S. Karlin, and H.M. Taylor, "A Second Course in Stochastic Processes", Academic Press, New York, 1981.

....See for example [RY90, M et82, Wil79, DM78, JS87, Pro92, vWW90, KS88, LS89] to name but a few. The theory associated with these definitions is usually called the general theory of processes. Constructions of BM x (R d ) and PP ( abound, especially that of PP ( which is much simpler (see [KT81] for instance) The book [Kni81] has many different constructions of BM x (R d ) but see also [SV79, Wil79, KS88, RY90] The proofs that BM 0 (R) has paths of infinite variation, along with many other esoteric properties are usually given in those books too. See in 52 particular [Nel67] ....

S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981.

....process is a Markovian process in continuous time which is completely described by the transition rates, P ; 1 , P ; and P ; Gamma1 . The replicationdeletion process with the approximations adopted above, and with a finite population of constant size N , is a birth and death process [25, 35, 36] that converges to a stationary distribution, defined by: dP =dt = 0 8 . The time axis can be transformed such that the probability for a replication deletion event becomes independent of the population average. This reactor time t is defined by d t : dt N (oe Gamma 1) N = dt ....

S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981.

....1] with infinitesimal generator 2y(1 Gamma y)D 2 Gamma d Gamma (d d 0 )y Delta D: 3. 1) These diffusions, called Jacobi diffusions with dimensions d and d 0 , have been well studied, particularly in relations to models in genetics, see for example Ethier Kurtz [8] Karlin Taylor [9] and Kimura [10] or more recently in financial models, see Delbaen Shirakawa [6] For other studies and motivations, including hypercontractivity, see Bakry [1] and Mazet [11] Some further results are given by the authors of this paper in [18] where the Jacobi diffusions are introduced via the ....

S. Karlin and H.M. Taylor. A second course in stochastic processes. Academic Press, New York, 1981.

.... fi fi fi fi t=s : Before defining a class of approximating martingales, observe that u (t; x) is the exact solution to the linear system u 1 (s; Delta) Au(s; Delta) where u(0; Delta) f( Delta) Note that this is just the Kolmogorov backwards equation for Y (see, e.g. chapter 14, Karlin and Taylor 1981). Proposition 6 Suppose that (u(s; Delta) 0 s t) is a set of real valued functions defined on S with the property that u(s; x) is continuously differentiable in s for all x 2 S, i.e. u 1 (s; x) is continuous in s for all x. Then M = M(s) 0 s t) is a P martingale for all , where M(s) ....

....of U Gamma M1 , where M1 is the almost sure limit of an appropriate martingale. Before defining a useful class of martingales, observe that u satisfies the linear system Au(x) Gamma h(x)u(x) Gammaf (x) 8x 2 IR; where Au(x) 4 = x)u 0 (x) 1 2 oe 2 (x)u 00 (x) On p. 191 of Karlin and Taylor (1981), similar systems are derived for related performance measures. Proposition 8 If h is bounded away from 0 and 1, and if u : IR IR is twice continuously differentiable with u 0 bounded, then M = M t : t 0) is a P martingale for all , where M t = e GammaV t u(X t ) Gamma u(X 0 ) Gamma Z ....

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Karlin, S. and H. M. Taylor. (1981). A Second Course in Stochastic Processes. Academic Press, Boston.

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S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes. Academic Press, 1981.

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Karlin, S. and Taylor, H. M. (1991) A Second Course in Stochastic Processes. Academic Press, Inc., New York.

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S. Karlin and H.M. Taylor. A Second Course in Stochastic Processes. Academic, New York, 1981.

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S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981.

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S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981.

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Samuel Karlin and Howard M. Taylor. A second course in stochastic processes. Academic Press, New York, 1981.

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S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes. Academic Press, 1981.

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Karlin, Samuel and Howard Taylor (1981), A Second Course in Stochastic Processes, Academic Press, New York.

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Karlin, Samuel, and Howard M. Taylor, 1981, A Second Course in Stochastic Processes, Academic Press, Inc., New York, 1981.

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S. Karlin, H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York 1975

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