Transition probability

A transition probability matrix is called doubly stochastic if the columns sum to one as well as the rows. Formally, P = || Pij || is doubly stochastic if. P i j ≥ 0 and ∑ k P i k = ∑ k P k j = 1 for all i, j. Consider a doubly stochastic transition probability matrix on the N states 0, 1, …, N − 1.

29 Sept 2021 ... In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on \mathbb{Z} in ...P ( X t + 1 = j | X t = i) = p i, j. are independent of t where Pi,j is the probability, given the system is in state i at time t, it will be in state j at time t + 1. The transition probabilities are expressed by an m × m matrix called the transition probability matrix. The transition probability is defined as:I want to essentially create a total transition probability where for every unique page— I get a table/matrix which has a transition probability for every single possible page. ... To build a transition matrix, it is often easy to first build a matrix of counts. The counts can then be divided to produce transition probabilities.Periodicity is a class property. This means that, if one of the states in an irreducible Markov Chain is aperiodic, say, then all the remaining states are also aperiodic. Since, p(1) aa > 0 p a a ( 1) > 0, by the definition of periodicity, state a is aperiodic.The transportation channel explains how people and goods get from place to place. Check out this collection of transportation articles. Advertisement Many of us take public transportation or fly in airplanes on a regular basis, but have you...If the probability of bit transition is only dependent on the original bit value, but independent of the position (i.e. P(xy|ab) == P(yx|ba), then you can simply block-multiply a kernel of transition probabilities: Let x be a 2x2 matrix such that x[i,j] is the probability of observing bit j given the truth i.I.e.: x = [[a, b] [c, d]]A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now."A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete ...

Sorted by: 1. They're just saying that the probability of ending in state j j, given that you start in state i i is the element in the i i th row and j j th column of the matrix. For example, if you start in state 3 3, the probability of transitioning to state 7 7 is the element in the 3rd row, and 7th column of the matrix: p37 p 37. Share. Cite.from assigns probability π(x) to x. The function p(x) is known and Z is a constant which normalizes it to make it a probability distribution. Z may be unknown. Let q(x,y) be some transition function for a Markov chain with state space S. If S is discrete then q(x,y) is a transition probability, while if S is continuous it is a transition ...In Table 4, we estimate the first order transition probability matrices for two different twelve-month periods between January 2001 and December 2004, in order to determine the effect of calendar time on transition probabilities. The first matrix is based on a sample of customers who were on the books during the period January-December 2001 ...Probability/risk #of events that occurred in a time period #of people followed for that time period 0-1 Rate #of events that occurred in a time period Total time period experienced by all subjects followed 0to Relativerisk Probability of outcome in exposed Probability of outcome in unexposed 0to Odds Probability of outcome 1−Probability of ...The transition probability under the action of a perturbation is given, in the first approximation, by the well-known formulae of perturbation theory (QM, §42). Let the initial and final states of the emitting system belong to the discrete spectrum. † Then the probability (per unit time) of the transitioni→fwith emission of a photon isA transition matrix consists of a square matrix that gives the probabilities of different states going from one to another. With a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications. Consider the table showing the purchasing patterns involving different cereals.We will refer to \(\rho\) as the risk of death for healthy patients. As there are only two possible transitions out of health, the probability that a transition out of the health state is an \(h \rightarrow i\) transition is \(1-\rho\).. The mean time of exit from the healthy state (i.e. mean progression-free survival time) is a biased measure in the presence of right censoring [].

Each transition adds some Gaussian noise to the previous one; it makes sense for the limiting distribution (if there is one) to be completely Gaussian. ... Can we use some "contraction" property of the transition probability to show it's getting closer and closer to Gaussian ? $\endgroup$Here, transition probability describes the likelihood of a certain transition between possible states at a given time. Additional subject-related variables can be incorporated by introducing a regression component into intensity matrix Q, such as demographic characteristics and functional assessments. Mean sojourn time refers to the average ...High probability here refers to different things: the book/professor might be not very clear about it.. The perturbation is weak and the transition rate is small - these are among the underlying assumptions of the derivation. Fermi Golden rule certainly fails when probabilities are close to $1$ - in this case it is more appropriate to discuss Rabi oscillations.$|c_i(t)|^2$ is interpreted as transition probability in perturbative treatments, such as Fermi golden rule. That is, we are still looking at the states of the unperturbed Hamiltonian, and what interests us is how the population of these states changes with time (due to the presence of the perturbation.). When perturbation is strong, i.e., cannot be considered perturbatively, as, e.g., in the ...

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Transition probabilities The probabilities of transition of a Markov chain $ \xi ( t) $ from a state $ i $ into a state $ j $ in a time interval $ [ s, t] $: $$ p _ {ij} ( s, t) = …PublicRoutes tells you how to get from point A to point B using public transportation. PublicRoutes tells you how to get from point A to point B using public transportation. Just type in the start and end addresses and the site spits out de...Apr 27, 2017 · The probability that the system goes to state i + 1 i + 1 is 3−i 3 3 − i 3 because this is the probability that one selects a ball from the right box. For example, if the system is in state 1 1 then there is only two possible transitions, as shown below. The system can go to state 2 2 (with probability 23 2 3) or to state 0 0 (with ... and a transition probability kernel (that gives the probabilities that a state, at time n+1, succeeds to another, at time n, for any pair of states) denoted. With the previous two objects known, the full (probabilistic) dynamic of the process is well defined. Indeed, the probability of any realisation of the process can then be computed in a ...If at a hotel, he returns to the airport with probability 3=4 or goes to the other hotel with probability 1=4. (a) Find the transition probability matrix for this Markov chain. (b) Suppose the driver begins at the airport at time 0. Find the probability that he is back at the airport at time 2. (c) Suppose the driver begins at the airport at ...

The sensitivity of the spectrometer is crucial. So too is the concentration of the absorbing or emitting species. However, our interest in the remainder of this chapter is with the intrinsic transition probability, i.e. the part that is determined solely by the specific properties of the molecule. The key to understanding this is the concept of ...We then look up into the Markov transition matrix to get the probability that a value from bin 2 transitions into bin 1; This value is 10.7%, hence M[1,6] = 10.7%; The transition that happens between timestep x[1] and x[6] has a 10.7% chance of happening when looking at the whole signal. Let's now plot the transition field we just computed:Transition probability is the probability of someone in one role (or state) transitioning to another role (or state) within some fixed period of time. The year is the typical unit of time but as with other metrics that depend on events with a lower frequency, I recommend you look at longer periods (e.g. 2 years) too.Mar 4, 2014 · We show that if [Inline formula] is a transition probability tensor, then solutions of this [Inline formula]-eigenvalue problem exist. When [Inline formula] is irreducible, all the entries of ...Mar 1, 2006 · 1.. IntroductionIn Part 1 of the paper Du and Yeung (2004), we have presented a new condition monitoring method: fuzzy transition probability (FTP).The new method is based on a combination of fuzzy set and Markov process. The fuzzy set is used to describe the ambiguous states of a monitored process (e.g., in machining tool wear may be …The energy of the photon E E E absorbed/released during the transition is equal to the energy change Δ E \Delta E ΔE of the electron. What is state transition probability? The state transition probability matrix of a Markov chain gives the probabilities of transitioning from one state to another in a single time unit.$|c_i(t)|^2$ is interpreted as transition probability in perturbative treatments, such as Fermi golden rule. That is, we are still looking at the states of the unperturbed Hamiltonian, and what interests us is how the population of these states changes with time (due to the presence of the perturbation.). When perturbation is strong, i.e., cannot be considered perturbatively, as, e.g., in the ...State Transition Matrix For a Markov state s and successor state s0, the state transition probability is de ned by P ss0= P S t+1 = s 0jS t = s State transition matrix Pde nes transition probabilities from all states s to all successor states s0, to P = from 2 6 4 P 11::: P 1n... P n1::: P nn 3 7 5 where each row of the matrix sums to 1.Oct 15, 2015 · 1 Answer. The best way to present transition probabilities is in a transition matrix where T (i,j) is the probability of Ti going to Tj. Let's start with your data: import pandas as pd import numpy as np np.random.seed (5) strings=list ('ABC') events= [strings [i] for i in np.random.randint (0,3,20)] groups= [1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2 ... The estimation of the transition probability between statuses at the account level helps to avoid the lack of memory in the MDP approach. The key question is which approach gives more accurate results: multinomial logistic regression or multistage decision tree with binary logistic regressions. ...

In terms of probability, this means that, there exists two integers m > 0, n > 0 m > 0, n > 0 such that p(m) ij > 0 p i j ( m) > 0 and p(n) ji > 0 p j i ( n) > 0. If all the states in the Markov Chain belong to one closed communicating class, then the chain is called an irreducible Markov chain. Irreducibility is a property of the chain.

where A ki is the atomic transition probability and N k the number per unit volume (number density) of excited atoms in the upper (initial) level k. For a homogeneous light source of length l and for the optically thin case, where all radiation escapes, the total emitted line intensity (SI quantity: radiance) isCΣ is the cost of transmitting an atomic message: . •. P is the transition probability function. P ( s ′| s, a) is the probability of moving from state s ∈ S to state s ′∈ S when the agents perform actions given by the vector a, respectively. This transition model is stationary, i.e., it is independent of time. In chemistry and physics, selection rules define the transition probability from one eigenstate to another eigenstate. In this topic, we are going to discuss the transition moment, which is the key to …Mar 6, 2012 · Transition probability It is not essential that exposure of a compound to ultraviolet or visible light must always gives to an electronic transition. On the other hand, the probability of a particular electronic transition has found to depend € d upon the value of molar extinction coefficient and certain other factors. According transitions ...Dec 20, 2011 · Transition Probability Geostatistical Software (T-PROGS) is a set of FORTRAN computer pro-grams that implements a transition probability/Markov approach to geostatistical analysis and simulation of spatial distributions of categorical variables (e.g., geologic units, facies). Im-Here the transition probability from state ito state jafter t+sunits is given X k P(t) ik P (s) kj = P (t+s) ij, which means (1.1.2) is valid. Naturally P = I. Just as in the case of Markov chains it is helpful to explicitly describe the structure of the underlying probability space Ω of a continuous time Markov chain. Here Ω is the space of ...The probability formalization of a stochastic process is now well known. In the present case the initial distribution and the transition probabilities are used to define a probability measure in the space of all functions x(i), where tç^to, and x(i) is a function which takes on values in X. For example, to the

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State space and transition probability of Markov Chain. 0. Confused with the definition of hitting time (Markov chains) 2. First time two independent Markov chains reach same state. 1. Probability distribution of time-integral of a two-state continuous-time Markov process. Hot Network QuestionsApr 1, 2021 · As depicted in Fig. 5, Fig. 6, it can be seen that the two competing Markov-switching models, namely, the time-varying transition probability and the constant transition probability models have its own superiority. It is also worth noting that even though the time-varying transition probability models ranked at the top of MCS ranking but the ...A stationary probability vector π is defined as a distribution, written as a row vector, that does not change under application of the transition matrix; that is, it is defined as a probability distribution on the set {1, …, n} which is also a row eigenvector of the probability matrix, associated with eigenvalue 1: The transition dipole moment integral and its relationship to the absorption coefficient and transition probability can be derived from the time-dependent Schrödinger equation. Here we only want to introduce the concept of the transition dipole moment and use it to obtain selection rules and relative transition probabilities for the particle ...Math; Statistics and Probability; Statistics and Probability questions and answers; Consider the Markov chain whose transition probability matrix is given by 0 1 2 3 ...TECHNICAL BRIEF • TRANSITION DENSITY 2 Figure 2. Area under the left extreme of the probability distribution function is the probability of an event occurring to the left of that limit. Figure 3. When the transition density is less than 1, we must find a limit bounding an area which is larger, to compensate for the bits with no transition.Assuming that there are no absorbing states and using the Strong Markov Property i want to show that (Zm)m≥0 ( Z m) m ≥ 0 is a Markov chain and why the …Transition state theory is an equilibrium formulation of chemical reaction rates that originally comes from classical gas-phase reaction kinetics. ... (E^f_a - E^r_a = \Delta G^0_{rxn}\). P i refers to the population or probability of occupying the reactant or product state. The primary assumptions of TST is that the transition state is well ...The transition-probability model proposed, in its original form, 44 that there were two phases that regulated the interdivision time distribution of cells. There was a probabilistic phase and a constant phase. The probabilistic phase was thought to be associated with the variable G1 phase, while the constant phase was associated with the more ... the probability of being in a transient state after N steps is at most 1 - e ; the probability of being in a transient state after 2N steps is at most H1-eL2; the probability of being in a transient state after 3N steps is at most H1-eL3; etc. Since H1-eLn fi 0 as n fi ¥ , the probability of the(a) Compute its transition probability. (b) Compute the two-step transition probability. (c) What is the probability it will rain on Wednesday given that it did not rain on Sunday or Monday?Sep 28, 2023 · The transition kernel K t is defined by some measurability conditions and by the fact that, for every measurable Borel set A and every (bounded) measurable function u, E ( u ( X t): X t + 1 ∈ A) = E ( u ( X t) K t ( X t, A)). Hence, each K t ( ⋅, A) is defined only up to sets of measure zero for the distribution of X t, in the following ... ….

Transition probability geostatistical is a geostatistical method to simulate hydrofacies using sequential indicator simulation by replacing the semivariogram function with a transition probability model. Geological statistics information such as the proportion of geological types, average length, and transition trend among geological types, are ...Dec 20, 2011 · Transition Probability Geostatistical Software (T-PROGS) is a set of FORTRAN computer pro-grams that implements a transition probability/Markov approach to geostatistical analysis and simulation of spatial distributions of categorical variables (e.g., geologic units, facies). Im-We first measured the actual transition probabilities between actions to serve as a “ground truth” against which to compare people’s perceptions. We computed these ground truth transition probabilities using five different datasets. In study 1, we analyzed actions in movies, using movie scripts from IMSDb.com.The figure below depicts a latent transition model with four indicators. τ jc as the response probability and α 2|1 as the intercept/threshold for the multinomial logistic. 1. Newsom (2015), p. 276 . In addition to the response probabilities, transition probabilities are estimated represents the probabilityLet {α i: i = 1,2, . . .} be a probability distribution, and consider the Markov chain whose transition probability matrix isWhat condition on the probability distribution {α i: i = 1,2, . . .} is necessary and sufficient in order that a limiting distribution exist, and what is this limiting distribution?Assume α 1 > 0 and α 2 > 0 so that the chain is aperiodic.The above equation shows that the probability of the electron being in the initial state decays exponentially with time because the electron is likely to make a transition to another state. The probability decay rate is given by, n k k n n k n k k n n k H H 2 ˆ 2 2 ˆ 2 Note that the probability decay rate consists of two parts.If this were a small perturbation, then I would simply use first-order perturbation theory to calculate the transition probability. However, in my case, the perturbation is not small . Therefore, first order approximations are not valid, and I would have to use the more general form given below:One-step Transition Probability p ji(n) = ProbfX n+1 = jjX n = ig is the probability that the process is in state j at time n + 1 given that the process was in state i at time n. For each state, p ji satis es X1 j=1 p ji = 1 & p ji 0: I The above summation means the process at state i must transfer to j or stay in i during the next time ...The binary symmetric channel (BSC) with crossover probability p, shown in Fig. 6, models a simple channel with a binary input and a binary output which generally conveys its input faithfully, but with probability p flips the input. Formally, the BSC has input and output alphabets χ = = {0,1} and. FIGURE 6 Binary symmetric channel. Transition probability, Jan 1, 2021 · 一、基本概念 转移概率(Transition Probability) 从一种健康状态转变为另一种健康状态的概率（状态转换模型，state-transition model） 发生事件的概率（离散事件模拟，discrete-event simulations） 二、获取转移概率的方法 从现存的单个研究中获取数据 从现存的多个研究中合成数据：Meta分析、混合处理比较（Mixed ... , Transition Probability between states (T) If we are in the state S₂, the probability of staying put in S₂ is 0.1, transitioning to state S₁ is 0, and transitioning to state S₃ is 0.9 (as evident from the second row in the matrix)., Sep 1, 2017 · Conclusions. There is limited formal guidance available on the estimation of transition probabilities for use in decision-analytic models. Given the increasing importance of cost-effectiveness analysis in the decision-making processes of HTA bodies and other medical decision-makers, there is a need for additional guidance to inform a more consistent approach to decision-analytic modeling. , All statistical analyses were conducted in RStudio v1.3.1073 (R Core Team 2020).A Kaplan-Meier model was used to analyse the probability of COTS in experiment 1 transitioning at each time point (R-package "survival" (Therneau 2020)).The probability of juvenile COTS transitioning to coral at the end of the second experiment, and the survival of COTS under the different treatments, was ..., Transition probabilities The probabilities of transition of a Markov chain $ \xi ( t) $ from a state $ i $ into a state $ j $ in a time interval $ [ s, t] $: $$ p _ {ij} ( s, t) = …, Transition Probabilities. The one-step transition probability is the probability of transitioning from one state to another in a single step. The Markov chain is said to be time homogeneous if the transition probabilities from one state to another are independent of time index . The transition probability matrix, , is the matrix consisting of ... , In Fig. 8, we have plotted the transition probability Q as a function of the period of oscillation t at different the SEPC \( \alpha \) (Fig. 6a), the MFCF \( \omega_{\text{c}} \) (Fig. 8b) and the electric field F (Fig. 8c). The probability Q in Fig. 8 periodically oscillates with the oscillation period t. This phenomenon originates from Eq., Feb 5, 2004 · This formula has direct application to the process of transforming probability density functions::: Suppose X is a random variable whose probability density function is f(x). By de nition: P(a 6 X < b) = Z b a f(x)dx (11:2) Any function of a random variable is itself a random variable and, if y is taken as some, excluded. However, if one specifies all transition matrices p(t) in 0 < t ≤ t 0 for some t 0 > 0, all other transition probabilities may be constructed from these. These transition probability matrices should be chosen to satisfy the Chapman-Kolmogorov equation, which states that: P ij(t+s) = X k P ik(t)P kj(s), Apr 1, 2021 · As depicted in Fig. 5, Fig. 6, it can be seen that the two competing Markov-switching models, namely, the time-varying transition probability and the constant transition probability models have its own superiority. It is also worth noting that even though the time-varying transition probability models ranked at the top of MCS ranking but the ..., The transition probability for the two-photon process has been analyzed in detail by Breit and Teller [3] and Shapiro and Breit [4]. We have adopted variational equivalent of the formula given by equation (6.2) due to Breit and Teller [3] for transition to a two-photon excited state via an intermediate virtual state lying at half of the two ..., P (new=C | old=D) P (new=D | old=D) I can do it in a manual way, summing up all the values when each transition happens and dividing by the number of rows, but I was wondering if there's a built-in function in R that calculates those probabilities or at least helps to fasten calculating those probabilities., which possesses a transition probability density pt(x,y). To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which even allows us to handle the case 0 <α≤1and b=0, i.e. when the gradient part of the generator is not dominated by the jump part. Résumé., nn a transition probability matrix A, each a ij represent-ing the probability of moving from stateP i to state j, s.t. n j=1 a ij =1 8i p =p 1;p 2;:::;p N an initial probability distribution over states. p i is the probability that the Markov chain will start in state i. Some states jmay have p j =0, meaning that they cannot be initial states ..., This discrete-time Markov decision process M = ( S, A, T, P t, R t) consists of a Markov chain with some extra structure: S is a finite set of states. A = ⋃ s ∈ S A s, where A s is a finite set of actions available for state s. T is the (countable cardinality) index set representing time. ∀ t ∈ T, P t: ( S × A) × S → [ 0, 1] is a ..., Gauss kernel, which is the transition probability function for Brownian motion: (4) P(W t+s2dyjW s= x) = p t(x;y)dy= 1 p 2ˇt expf (y x)2=2tgdy: This equation follows directly from properties (3)-(4) in the definition of a standard Brow-nian motion, and the definition of the normal distribution. The function p, The transition probability λ is also called the decay probability or decay constant and is related to the mean lifetime τ of the state by λ = 1/τ. The general form of Fermi's golden rule can apply to atomic transitions, nuclear decay, scattering ... a large variety of physical transitions. A transition will proceed more rapidly if the ... , Sep 28, 2023 · The transition kernel K t is defined by some measurability conditions and by the fact that, for every measurable Borel set A and every (bounded) measurable function u, E ( u ( X t): X t + 1 ∈ A) = E ( u ( X t) K t ( X t, A)). Hence, each K t ( ⋅, A) is defined only up to sets of measure zero for the distribution of X t, in the following ..., Abstract The Data Center on Atomic Transition Probabilities at the U.S. National Institute of Standards and Technology (NIST), formerly the National Bureau of Standards (NBS), has critically evaluated and compiled atomic transition probability data since 1962 and has published tables containing data for about 39,000 transitions of the 28 lightest elements, hydrogen through nickel., Detuning in Rabi oscillations. with ΩR = [Δ2 +ν2/ℏ2]1/2 Ω R = [ Δ 2 + ν 2 / ℏ 2] 1 / 2 and ν =< e|V^0|g > ν =< e | V ^ 0 | g >. The plot of Probability vs time for various values of Δ Δ is given. The question is when detuning factor Δ Δ is non-zero i.e, Δ Δ increases the amplitude of the probability decreases and the time ..., Picture showing Transition probabilities and Emission Probabilities. We calculate the prior probabilities. P(S)=0.67 and P(R)=0.33. Now, let’s say for three days Bob is Happy, Grumpy, Happy then ..., In Fig. 8, we have plotted the transition probability Q as a function of the period of oscillation t at different the SEPC \( \alpha \) (Fig. 6a), the MFCF \( \omega_{\text{c}} \) (Fig. 8b) and the electric field F (Fig. 8c). The probability Q in Fig. 8 periodically oscillates with the oscillation period t. This phenomenon originates from Eq., Other articles where transition probability is discussed: probability theory: Markovian processes: …given X(t) is called the transition probability of the process. If this conditional distribution does not depend on t, the process is said to have "stationary" transition probabilities. A Markov process with stationary transition probabilities may or may not be a stationary process in the ..., For computing the transition probabilities for a given STG, we need to know the probability distribution for the input nodes. The input probability can be ..., , An Introduction to Stochastic Modeling (4th Edition) Edit edition Solutions for Chapter 3.2 Problem 6E: A Markov chain X0,X1,X2, . . . has the transition probability matrixand initial distribution p0 = 0.5 and p1 = 0.5. Determine the probabilities Pr{X2 = 0} and Pr{X3 = 0}. …, Feb 26, 2021 · We first measured the actual transition probabilities between actions to serve as a “ground truth” against which to compare people’s perceptions. We computed these ground truth transition probabilities using five different datasets. In study 1, we analyzed actions in movies, using movie scripts from IMSDb.com. , Atomic Transition Probabilities and Lifetimes 1105 quantum state i is (1) where thus Aki is introduced as the probability, per unit time, that spon­ taneous emission takes place. The radiative lifetime of an excited atomic state k follows from the consideration that this state decays radiatively, in the absence of absorp­, Consider the following transition probability graph: This figure depicts a Markov chain with three possible states. The possible states are S_1, S_2, and S_3, which are depicted as a row of circles on the middle of the diagram and placed from left to right in this order. At the upper part of the diagram, there are self-loops within S_1, S_2, and S_3, which are circular arrows with both the ..., The transition probability matrix \( P_t \) of \( \bs{X} \) corresponding to \( t \in [0, \infty) \) is \[ P_t(x, y) = \P(X_t = y \mid X_0 = x), \quad (x, y) \in S^2 \] In particular, …, Background Multi-state models are being increasingly used to capture complex disease pathways. The convenient formula of the exponential multi-state model can facilitate a quick and accessible understanding of the data. However, assuming time constant transition rates is not always plausible. On the other hand, obtaining predictions from a fitted model with time-dependent transitions can be ..., 1 Answer. Let pi p i be the probability that the process is eventually absorbed by s1 s 1 after starting at si s i. Then p1 = 1 p 1 = 1, p5 = 0 p 5 = 0 and. p2 p3 p4 = 0.7p1 + 0.3p3, = 0.5p2 + 0.5p4, = 0.65p3 + 0.35p5. p 2 = 0.7 p 1 + 0.3 p 3, p 3 = 0.5 p 2 + 0.5 p 4, p 4 = 0.65 p 3 + 0.35 p 5. This system of three linear equations in three ..., More generally, suppose that \( \bs{X} \) is a Markov chain with state space \( S \) and transition probability matrix \( P \). The last two theorems can be used to test whether an irreducible equivalence class \( C \) is recurrent or transient.