Transition probability

The purpose of the present vignette is to demonstrate the visualisation capacities of mstate, using both base R graphics and the ggplot2 package (Wickham 2016). To do so, we will use the dataset used to illustrate competing risks analyses in Section 3 of the Tutorial by Putter, Fiocco, and Geskus (2007) . The dataset is available in mstate ...

Transition probability. The following code provides another solution about Markov transition matrix order 1. Your data can be list of integers, list of strings, or a string. The negative think is that this solution -most likely- requires time and memory. generates 1000 integers in order to train the Markov transition matrix to a dataset.

reverse of Transition Probability Density function. Given 2 distributions with the probability density functions p(x) p ( x) and q(y) q ( y), and their transition probability density function T(y, x) T ( y, x), we have. In which situation, there would exist a "reverse of transition probability density function" R(y, x) R ( y, x) such that.

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 isThe transition probability (a.k.a. Einstein coefficient, A-coefficient, oscillator strength, gf-value) is a temperature independent property representing the spontaneous emission rate in a two-level energy model.Limit Behavior of Transition Probability Matrix. 0. Find probability of markov chain ended in state $0$. 0. Markov chain equivalence class definition. 1. Stationary distribution of a DTMC that has recurrent and transient states. Hot Network Questions Does Fide/Elo rating fade over time?2. People often consider square matrices with non-negative entries and row sums ≤ 1 ≤ 1 in the context of Markov chains. They are called sub-stochastic. The usual convention is the missing mass 1 − ∑[ 1 − ∑ [ entries in row i] i] corresponds to the probability that the Markov chain is "killed" and sent to an imaginary absorbing ...$\begingroup$ @Wayne: (+1) You raise a good point. I have assumed that each row is an independent run of the Markov chain and so we are seeking the transition probability estimates form these chains run in parallel. But, even if this were a chain that, say, wrapped from one end of a row down to the beginning of the next, the estimates …

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}. …probability theory. Probability theory - Markov Processes, Random Variables, Probability Distributions: A stochastic process is called Markovian (after the Russian mathematician Andrey Andreyevich Markov) if at any time t the conditional probability of an arbitrary future event given the entire past of the process—i.e., given X (s) for all s ...$\begingroup$ While that source does not give the result in precisely those words, it does show on p 34 that an irreducible chain with an aperiodic state is regular, which is a stronger result, because if an entry on the main diagonal of the chain's transition matrix is positive, then the corresponding state must be aperiodic. $\endgroup$tabulated here. Transition probabilities are given in units of s 1. Lower level and Upper level indicate the classification given for the transition. Ref. and A ki Ref. indicate the references for the wave-length measurement and transition probability, respectively. The list of references for each ionization stage is located atConsider 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 ...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.The transition probability from fair to fair is highest at around 55 percent for 60-70 year olds, and the transition probability from Poor to Poor is highest at around 50 percent for 80 year olds. Again this persistence of remaining in worse and worse health states as one ages is consistent with the biological aging process and the ...

In this diagram, there are three possible states 1 1, 2 2, and 3 3, and the arrows from each state to other states show the transition probabilities pij p i j. When there is no arrow from state i i to state j j, it means that pij = 0 p i j = 0 . Figure 11.7 - A state transition diagram. Example. Consider the Markov chain shown in Figure 11.7.Explicitly give the transition probability matrix \( P \). Suppose that the initial distribution is the uniform distribution on \( \{000, 001, 101, 100\} \). Find the probability density function of \( X_2 \). Answer. For the matrix and vector below, we use the ordered state space \( S = (000, 001, 101, 110, 010, 011, 111, 101 ) \).Apr 24, 2022 · A standard Brownian motion is a random process X = {Xt: t ∈ [0, ∞)} with state space R that satisfies the following properties: X0 = 0 (with probability 1). X has stationary increments. That is, for s, t ∈ [0, ∞) with s < t, the distribution of Xt − Xs is the same as the distribution of Xt − s. X has independent increments. A continuous-time Markov chain on the nonnegative integers can be defined in a number of ways. One way is through the infinitesimal change in its probability transition function …Abstract. In this paper, we propose and develop an iterative method to calculate a limiting probability distribution vector of a transition probability tensor [Inline formula] arising from a ...

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Transitional Probability. Transitional probability is a term primarily used in mathematics and is used to describe actions and reactions to what is called the "Markov Chain." This Markov Chain describes a random process that undergoes transitions from one state to another without the current state being dependent on past state, and likewise the ...Our transition probability results obtained in this work are compared with the accepted values from NIST [20] for all transitions and Opacity Project values for multiplet transitions [21]. Also we compare our results with the ones obtained by Tachiev and Fischer [22] for some transitions belonging to lower levels from MCHF calculations.Define the transition probability matrix P of the chain to be the XX matrix with entries p(i,j), that is, the matrix whose ith row consists of the transition probabilities p(i,j)for j …This is an exact expression for the Laplace transform of the transition probability P 0, 0 (t). Let the partial numerators in be a 1 = 1 and a n = −λ n− 2 μ n− 1, and the partial denominators b 1 = s + λ 0 and b n = s + λ n− 1 + μ n− 1 for n ≥ 2. Then becomesThe 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 ...6.3: The Kolmogorov Differential Equations. Let Pij(t) P i j ( t) be the probability that a Markov process { X(t); t ≥ 0 X ( t); t ≥ 0 } is in state j j at time t t given that X(0) = i X ( 0) = i, Pij(t) P i j ( t) is analogous to the nth n t h order transition probabilities Pnij P i j n for Markov chains.

The transition probability is the probability of sedimentary facies transitions at different lag distances within a three dimensional domain (Agterberg 1974). By incorporating facies spatial correlations, volumetric proportions, juxtapositional tendencies into a spatial continuity model, Carle and Fogg ( 1996 ) and Ritzi ( 2000 ) developed ...Taking the power of the transition matrix is a straightforward way to calculate what you want. But, given the simplicity of the states, for ending at state 2 2 after n n steps, you need to have odd parity and always alternate between states 1 and 2, i.e. each step is with 1/2 1 / 2 prob. So, P(Xn = 2|X0 = 1) = (1/2)n P ( X n = 2 | X 0 = 1 ...The transition probability matrix determines the probability that a pixel in one land use class will change to another class during the period analysed. The transition area matrix contains the number of pixels expected to change from one land use class to another over some time (Subedi et al., 2013). In our case, the land use maps of the area ...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:The new method, called the fuzzy transition probability (FTP), combines the transition probability (Markov process) as well as the fuzzy set. From a theoretical point of view, the new method uses the available information from the training samples to the maximum extent (finding both the transition probability and the fuzzy membership) and hence ...The transition probabilities are the probability of a tag occurring given the previous tag, for example, a verb will is most likely to be followed by another form of a verb like dance, so it will have a high probability. We can calculate this probability using the equation above, implemented below:Branch probability correlations range between 0.85 and 0.95, with 90% of correlations >0.9 (Supplementary Fig. 5d). Robustness to k , the number of neighbors for k- nearest neighbor graph constructionLearn more about markov chain, transition probability matrix Hi there I have time, speed and acceleration data for a car in three columns. I'm trying to generate a 2 dimensional transition probability matrix of velocity and acceleration.6.3: The Kolmogorov Differential Equations. Let Pij(t) P i j ( t) be the probability that a Markov process { X(t); t ≥ 0 X ( t); t ≥ 0 } is in state j j at time t t given that X(0) = i X ( 0) = i, Pij(t) P i j ( t) is analogous to the nth n t h order transition probabilities Pnij P i j n for Markov chains.Question: 1. Consider the Markov chain whose transition probability matrix is given by (a) Starting in state 2, determine the probability that the process is absorbed into state 0. (b) Starting in state 0, determine the mean time that the process spends in state 0 prior to absorption and the mean time that prior to absorption. (6m) [0.2 0.3 0 0 ...Rather, they are well-modelled by a Markov chain with the following transition probabilities: P = heads tails heads 0:51 0:49 tails 0:49 0:51 This shows that if you throw a Heads on your first toss, there is a very slightly higher chance of throwing heads on your second, and similarly for Tails. 3. Random walk on the line Suppose we perform a ...what are the probabilities of states 1 , 2 , and 4 in the stationary distribution of the Markov chain s shown in the image. The label to the left of an arrow gives the corresponding transition probability.

Oct 24, 2018 · As a transition probability, ASTP captures properties of the tendency to stay in active behaviors that cannot be captured by either the number of active breaks or the average active bout. Moreover, our results suggest ASTP provides information above and beyond a single measure of PA volume in older adults, as total daily PA declines and ...

the 'free' transition probability density function (pdf) is not sufficient; one is thus led to the more complicated task of determining transition functions in the pre-sence of preassigned absorbing boundaries, or first-passage-time densities for time-dependent boundaries (see, for instance, Daniels, H. E. [6], [7], Giorno, V. et al. [10 ...Jun 27, 2019 · The traditional Interacting Multiple Model (IMM) filters usually consider that the Transition Probability Matrix (TPM) is known, however, when the IMM is associated with time-varying or inaccurate ...Introduction. The transition probability is defined as the probability of particular spectroscopic transition to take place. When an atom or molecule absorbs a photon, the probability of an atom or molecule to transit from one energy level to another depends on two things: the nature of initial and final state wavefunctions and how strongly photons interact with an eigenstate.Multiple Step Transition Probabilities For any m ¥0, we de ne the m-step transition probability Pm i;j PrrX t m j |X t is: This is the probability that the chain moves from state i to state j in exactly m steps. If P pP i;jqdenotes the transition matrix, then the m-step transition matrix is given by pPm i;j q P m: 8/58by 6 coarse ratings instead of 21 fine ratings categories, before transforming the estimated coarse rating transition probabilities into fine rating transition probabilities. Table 1 shows the mapping between coarse and fine ratings. 1 EDF value is a probability of default measure provided by Moody's CreditEdge™.The transition probability P (q | p) is a characteristic of the algebraic structure of the observables. If the Hilbert space dimension does not equal two, we have S (L H) = S l i n (L H) and the transition probability becomes a characteristic of the even more basic structure of the quantum logic. 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)

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Mar 15, 2018 · In the case of D 2 lines, the transition probability varies in a nonlinear fashion with respect to magnetic eld. The non-dipole transitions (F = 3 to F 0 = 1) are also shown in gure 4, which have non zero transition probability with the application of magnetic eld. 4 Summary and Conclusions Magnetic eld induced transition probability for ...Transition Probabilities and Atomic Lifetimes. Wolfgang L. Wiese, in Encyclopedia of Physical Science and Technology (Third Edition), 2002 II Numerical Determinations. Transition probabilities for electric dipole transitions of neutral atoms typically span the range from about 10 9 s −1 for the strongest spectral lines at short wavelengths to 10 3 s …In Theorem 2 convergence is in fact in probability, i.e. the measure \(\mu \) of the set of initial conditions for which the distance of the transition probability to the invariant measure \(\mu \) after n steps is larger than \(\varepsilon \) converges to 0 for every \(\varepsilon >0\). It seems to be an open question if convergence even holds ...transition β,α -probability of given mutation in a unit of time" A random walk in this graph will generates a path; say AATTCA…. For each such path we can compute the probability of the path In this graph every path is possible (with different probability) but in general this does need to be true. Sep 16, 2022 · Transitional probability is a measure of how likely a symbol will appear, given a preceding or succeeding symbol. For a bigram AB, its forward transitional probability is the likelihood of B given A, and its backward transitional probability is the likelihood of A given B [Pelucci2009]. The measurement can be used to predict word or morpheme ...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 following code provides another solution about Markov transition matrix order 1. Your data can be list of integers, list of strings, or a string. The negative think is that this solution -most likely- requires time and memory. generates 1000 integers in order to train the Markov transition matrix to a dataset.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) is ….

Apr 1, 1976 · The transition probability P(ω,ϱ) is the spectrum of all the numbers |(x,y)| 2 taken over all such realizations. We derive properties of this straightforward generalization of the quantum mechanical transition probability and give, in some important cases, an explicit expression for this quantity.The state transition of the Markov chain can be categorized into six situations: (i) for and . This situation means that the test is passed. The state transition probability is presented as . (ii) for and . This situation means that the test is failed and the improvement action is accomplished so that the "consecutive- k successful run ...Markov chain with transition probabilities P(Y n+1 = jjY n =i)= pj pi P ji. The tran-sition probabilities for Y n are the same as those for X n, exactly when X n satisfies detailed balance! Therefore, the chain is statistically indistinguishable whether it is run forward or backward in time. Detailed balance is a very important concept in ...Apr 5, 2017 · As mentioned in the introduction, the “simple formula” is sometimes used instead to convert from transition rates to probabilities: p ij (t) = 1 − e −q ij t for i ≠ j, and p ii (t) = 1 − ∑ j ≠ i p ij (t) so that the rows sum to 1. 25 This ignores all the transitions except the one from i to j, so it is correct when i is a death ... doi: 10.1016/j.procs.2015.07.305 Building efficient probability transition matrix using machine learning from big data for personalized route prediction Xipeng Wang 1 , Yuan Ma 1 , Junru Di 1 , Yi L Murphey 1* and Shiqi Qiu 2†, Johannes Kristinsson 2 , Jason Meyer 2 , Finn Tseng 2 , Timothy Feldkamp 2 1 University of Michigan-Dearborn, USA. 2 Ford Motor …Draw the state transition diagram, with the probabilities for the transitions. b). Find the transient states and recurrent states. c). Is the Markov chain ...Besides, in general transition probability from every hidden state to terminal state is equal to 1. Diagram 4. Initial/Terminal state probability distribution diagram | Image by Author. In Diagram 4 you can see that when observation sequence starts most probable hidden state which emits first observation sequence symbol is hidden state F.17 Jul 2019 ... Transition Probability: The probability that the agent will move from one state to another is called transition probability. The Markov Property ... Transition probability, The modeled transition probability using the Embedded Markov Chain approach, Figure 5, successfully represents the observed data. Even though the transition rates at the first lag are not specified directly, the modeled transition probability fits the borehole data at the first lag in the vertical direction and AEM data in the horizontal direction., Static transition probability P 0 1 = P out=0 x P out=1 = P 0 x (1-P 0) Switching activity, P 0 1, has two components A static component –function of the logic topology A dynamic component –function of the timing behavior (glitching) NOR static transition probability = 3/4 x 1/4 = 3/16 , Figure 2: Illustration of the transition probability (density) appropriate for a Wiener process as a function of time. The initial distribution, set at time t= s= 0, is a delta function centered on w0= 1. and hence, in general E[W(t)W(s)] = min(t;s). Exercise: Prove it the old-fashioned way (i.e., by changing variables and integrating)!, 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 ..., Mar 15, 2018 · In the case of D 2 lines, the transition probability varies in a nonlinear fashion with respect to magnetic eld. The non-dipole transitions (F = 3 to F 0 = 1) are also shown in gure 4, which have non zero transition probability with the application of magnetic eld. 4 Summary and Conclusions Magnetic eld induced transition probability for ..., I would like to define a matrix of transition probabilities from edges with probabilities using define_transition from heemod. I am building a decision-tree where each edge represents a conditional probability of a decision. The end nodes in this tree are the edges that end with the .ts or .nts suffix., 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 ..., The Transition Probability Function P ij(t) Consider a continuous time Markov chain fX(t);t 0g. We are interested in the probability that in ttime units the process will be in state j, given that it is currently in state i P ij(t) = P(X(t+ s) = jjX(s) = i) This function is called the transition probability function of the process. , Statistics and Probability; Statistics and Probability questions and answers; 4. Let P and Q be transition probability matrices on states 1, ..., m, with respec- tive transition probabilities Pinj and Qi,j. Consider processes {Xn, n > 0} and {Yn, n >0} defined as follows: (a) Xo = 1. A coin that comes up heads with probability p is then flipped., Oct 24, 2018 · Methods. Participants of the Baltimore Longitudinal Study of Aging (n = 680, 50% male, aged 27–94 years) completed a clinical assessment and wore an Actiheart accelerometer.Transitions between active and sedentary states were modeled as a probability (Active-to-Sedentary Transition Probability [ASTP]) defined as the reciprocal …, 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:, People and Landslides - Humans contribute to the probability of landslides. Find out what activities make landslides more likely to occur. Advertisement Humans make landslides more likely through activities like deforestation, overgrazing, ..., However, the state transition probabilities are then also shown to cancel out exactly, so there is no requirement to know what the values are. State transition probabilities are irrelevant to probability ratios between identical trajectories where the policy varies but the environment does not. Which is the case for off-policy learning., probability theory. Probability theory - Markov Processes, Random Variables, Probability Distributions: A stochastic process is called Markovian (after the Russian mathematician Andrey Andreyevich Markov) if at any time t the conditional probability of an arbitrary future event given the entire past of the process—i.e., given X (s) for all s ..., 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., This divergence is telling us that there is a finite probability rate for the transition, so the likelihood of transition is proportional to time elapsed. Therefore, we should divide by \(t\) to get the transition rate. To get the quantitative result, we need to evaluate the weight of the \(\delta\) function term. We use the standard result, Besides, in general transition probability from every hidden state to terminal state is equal to 1. Diagram 4. Initial/Terminal state probability distribution diagram | Image by Author. In Diagram 4 you can see that when observation sequence starts most probable hidden state which emits first observation sequence symbol is hidden state F., Your expression is a result valid to first order in the perturbation. For long times restricting to first order is a poor approximation and one should include higher order terms. A sign that keeping only the first order term is poor is precisely that the transition probability becomes unphysically greater than 1., In Theorem 2 convergence is in fact in probability, i.e. the measure \(\mu \) of the set of initial conditions for which the distance of the transition probability to the invariant measure \(\mu \) after n steps is larger than \(\varepsilon \) converges to 0 for every \(\varepsilon >0\). It seems to be an open question if convergence even holds ..., TheGibbs Samplingalgorithm constructs a transition kernel K by sampling from the conditionals of the target (posterior) distribution. To provide a speci c example, consider a bivariate distribution p(y 1;y 2). Further, apply the transition kernel That is, if you are currently at (x 1;x 2), then the probability that you will be at (y 1;y, Rotational transitions; A selection rule describes how the probability of transitioning from one level to another cannot be zero.It has two sub-pieces: a gross selection rule and a specific selection rule.A gross selection rule illustrates characteristic requirements for atoms or molecules to display a spectrum of a given kind, such as an IR spectroscopy or a microwave spectroscopy., Exercise 16 Consider a Markov chain with state space S = f1,2,3gand transition matrix P = 2 4 0.2 0.4 0.4 0.1 0.5 0.4 0.6 0.3 0.1 3 5 Compute the probability that, in the long run, the chain is in state 1 (does the answer depend on the initial state X0?). Solve this problem in two different ways: (a) by computing the matrix Pn and letting n !¥;, 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 3 (Radiationless decay - loss of energy as heat) The transitions labeled with the number (3) in Figure 3.2.4 3.2. 4 are known as radiationless decay or external conversion. These generally correspond to the loss of energy as heat to surrounding solvent or other solute molecules. S1 = S0 + heat S 1 = S 0 + h e a t., 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 system is memoryless. A Markov Chain is a sequence of time-discrete transitions under the Markov Property with a finite state space. In this article, we will discuss The Chapman-Kolmogorov Equations and how these are used to calculate the multi-step transition probabilities for a given Markov Chain., 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., stimulated absorption: light induces a transition from 0 to 1 stimulated emission: light induces a transition from 1 to 0 In the emission process, the emitted photon is identical to the photon that caused the emission! Stimulated transitions: likelihood depends on the number of photons around A collection of two-level atoms, We applied a multistate Markov model to estimate the annual transition probabilities ... The annual transition probability from none-to-mild, mild-to-moderate and ..., 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:, This is an exact expression for the Laplace transform of the transition probability P 0, 0 (t). Let the partial numerators in be a 1 = 1 and a n = −λ n− 2 μ n− 1, and the partial denominators b 1 = s + λ 0 and b n = s + λ n− 1 + μ n− 1 for n ≥ 2. Then becomes, Let {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 transition probabilities $ p _ {ij} ( t) $ for a Markov chain with discrete time are determined by the values of $ p _ {ij} ( 1) $, $ i, j \in S $; for any $ t > 0 $, $ i \in S $, ... = 1 $, i.e. the path of $ \xi ( t) $" tends to infinity in a finite time with probability 1" (see also Branching processes, regularity of). References [1] K.L ...