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Absorbing Markov chain

An absorbing Markov chain is a stochastic process modeled as a Markov chain with at least one absorbing state—a state from which the process cannot transition away (i.e., the transition probability to itself is 1)—and the property that from every state in the chain, it is possible to reach an absorbing state in a finite number of steps.^[1]^[2] This structure ensures that the chain will eventually be "absorbed" into one of the absorbing states with probability 1, terminating the transient behavior.^[2]^[3] In an absorbing Markov chain, states are classified as either absorbing or transient: absorbing states trap the process indefinitely, while transient states are non-absorbing and will be left behind as the chain progresses toward absorption.^[1]^[3] The transition matrix of such a chain can be expressed in canonical form by partitioning states into transient (t states) and absorbing (r states), yielding the block matrix

P = \begin{pmatrix} Q & R \\ 0 & I \end{pmatrix},

where Q is the t \times t submatrix of transient-to-transient transitions, R is the t \times r submatrix of transient-to-absorbing transitions, and I is the r \times r identity matrix for absorbing states.^[1]^[2] This form highlights the chain's reductive nature, distinguishing it from irreducible Markov chains where no absorption occurs.^[2] Key analytical tools for absorbing Markov chains include the fundamental matrix N = (I - Q)^{-1}, whose (i,j)-entry represents the expected number of visits to transient state j starting from transient state i before absorption.^[1]^[3] Absorption probabilities are derived from the matrix B = NR, where the (i,j)-entry gives the probability of being absorbed in absorbing state j starting from transient state i.^[2]^[3] Additionally, the expected time to absorption from transient state i is the i-th entry of the vector N \mathbf{1}, where \mathbf{1} is a column vector of ones.^[1] These quantities enable precise computation of long-term behavior, such as in applications to queueing systems, genetics, and decision processes.^[2] A fundamental theorem guarantees that \lim_{n \to \infty} Q^n = 0, confirming certain absorption from any starting state.^[2]^[3]

Introduction

Definition

An absorbing Markov chain is a Markov chain defined on a finite or countably infinite state space that includes at least one absorbing state, with the additional property that from every state in the space, it is possible to reach at least one absorbing state.^[1] This structure distinguishes absorbing chains from general Markov chains; in finite state spaces, it ensures eventual termination in a stable state with probability 1, modeling processes like ruin in gambling or termination in queueing systems.^[4] While the definition applies to both finite and countably infinite state spaces, much of the analytical theory is developed for finite cases. In such a chain, absorption refers to the event where the process enters an absorbing state and remains there indefinitely with probability 1, as the transition probability from an absorbing state to itself is 1 and to all other states is 0.^[5] The concept originated in the context of Andrey Markov's foundational work on stochastic processes in the early 20th century (1906–1913), with absorbing variants gaining prominence in the 1910s through applications to gambling problems, such as the gambler's ruin scenario.^[6] For finite-state absorbing Markov chains, the process is absorbed almost surely—meaning with probability 1—if and only if there are no closed transient classes, a condition satisfied by the reachability requirement in the definition.^[5] This canonical structure can be analyzed by rearranging the states into absorbing and transient categories to highlight the chain's dynamics.^[1]

Key Properties

Absorbing Markov chains are distinguished from irreducible Markov chains by their inevitable termination in absorbing states, leading to a well-defined long-term behavior. Specifically, starting from any transient state, the chain converges to one of the absorbing states with probability 1, ensuring that the process is eventually absorbed and remains there indefinitely.^[1] In the limit, the occupancy probability of every transient state approaches zero, as all probability mass shifts entirely to the absorbing states.^[2] This contrasts sharply with irreducible chains, where the process continues indefinitely without absorption, maintaining positive long-run probabilities across all states. A key characteristic is that each transient state is visited only finitely many times almost surely, reflecting the transient nature of these states and the certainty of eventual absorption.^[7] This finite visitation ensures that the chain does not recurrently return to transient regions after leaving them, bounding the time spent in non-absorbing dynamics. In chains with multiple absorbing states, the process terminates in exactly one such state, with the absorption probabilities across all absorbing states summing to 1 for any starting transient state.^[1] To facilitate theoretical analysis, many treatments of absorbing Markov chains assume that the set of transient states forms a single communicating class, allowing reachability to absorbing states from all transients without multiple disconnected transient components.^[2]

State Classification

Absorbing States

In an absorbing Markov chain, an absorbing state is a state i such that the one-step transition probability p_{ii} = 1. This condition implies that, upon entering state i, the process remains there indefinitely with certainty, as the probability of transitioning to any other state j \neq i is zero.^[8]^[4] Absorbing states function as terminal points or "sinks" within the chain, where the stochastic process comes to a halt once reached. They represent scenarios in which the system stabilizes without further movement, such as winning or losing conditions in probabilistic models.^[1] To identify absorbing states in practice, examine the transition matrix P: the row corresponding to an absorbing state contains a 1 in the diagonal position p_{ii} and 0s in all off-diagonal positions of that row. This structure ensures no probability mass flows out of the state.^[4]^[1] Absorbing Markov chains may feature a single absorbing state, leading to one inevitable endpoint, or multiple absorbing states, which allow for various possible terminal outcomes depending on the path taken. For example, in a model of a tennis game at deuce, the states representing "Player A wins the game" and "Player B wins the game" are both absorbing, resulting in two distinct endpoints with absorption probabilities of approximately 0.6923 and 0.3077, respectively, when the probability of A winning a point is 0.6.^[1]

Transient States

In an absorbing Markov chain, a transient state j is a non-absorbing state from which there exists a path to at least one absorbing state with positive probability, and the probability of returning to j infinitely often, starting from j, is zero.^[9] This definition aligns with the general classification in Markov chain theory, where transience implies that the expected number of returns to the state is finite.^[2] From a transient state, the process will eventually depart and, upon reaching an absorbing state, never return to any transient state, including the original one, with probability 1.^[1] This behavior ensures that the chain spends only a finite amount of time in the set of transient states before absorption occurs.^[2] Transient states can be grouped into one or more communicating classes, where states within a class communicate with each other (mutual reachability with positive probability), but each such class leads to absorption in some absorbing state.^[9] These classes are inherently open, meaning there is a positive probability of exiting to another class or directly to absorption.^[2] In finite-state absorbing Markov chains, where every state can reach an absorbing state, all non-absorbing states are transient, as any recurrent non-absorbing class would be closed and prevent absorption.^[9] Consequently, the expected number of visits to a transient state before absorption remains finite.^[1]

Canonical Form

State Rearrangement

In absorbing Markov chains, state rearrangement serves to organize the state space in a manner that simplifies the structure of the transition matrix, thereby aiding analytical computations. By grouping all transient states together followed by the absorbing states, this procedure reveals a clear block form in the matrix, which is essential for deriving key properties such as absorption probabilities and expected absorption times. This canonical ordering is a foundational step in the theory, as it transforms an arbitrary transition matrix into a more tractable representation without altering the chain's probabilistic behavior.^[4] The rearrangement procedure entails permuting the states such that the t transient states occupy the initial indices 1 to t, while the r absorbing states are placed at indices t+1 to t+r. This permutation is achieved through a similarity transformation on the transition matrix, effectively reordering its rows and columns simultaneously to produce an upper-block triangular structure. For instance, in a chain with two transient states and three absorbing states, the states would be relabeled accordingly to position the transient ones first, ensuring the matrix blocks align for subsequent analysis.^[10] This approach is particularly effective for finite-state absorbing Markov chains, where the countable and finite state space permits explicit matrix construction and inversion. In cases of infinite state spaces, however, the rearrangement requires additional precautions, such as establishing conditions for almost sure absorption and employing operator-theoretic methods rather than finite matrices, to ensure the chain's behavior remains well-defined.^[11]

Transition Matrix Blocks

In the canonical form of the transition matrix for an absorbing Markov chain, the states are ordered such that the t transient states appear first, followed by the r absorbing states, where the total number of states is n = t + r. This rearrangement yields the block matrix

P = \begin{pmatrix} Q & R \\ 0 & I \end{pmatrix},

where Q is the t \times t matrix of transition probabilities among transient states, R is the t \times r matrix of transition probabilities from transient states to absorbing states, $0 is the r \times t zero matrix, and I is the r \times r identity matrix.^[12]^[13]^[2] The submatrix Q governs the dynamics within the transient subspace, with entries q_{ij} representing the probability of moving from transient state i to transient state j in one step.^[13]^[7] The submatrix R captures the direct one-step transition probabilities from each transient state to each absorbing state, quantifying the immediate absorption opportunities.^[12]^[2] The zero block $0 indicates that there are no transitions from absorbing states to transient states, maintaining the persistence of absorption.^[13]^[14] The block I reflects the defining property of absorbing states, ensuring that once the chain enters such a state, it remains there with probability 1, as the diagonal entries are 1 and off-diagonal entries within the absorbing block are 0.^[12]^[2] This canonical form assumes a finite state space and that the chain is absorbing, meaning every transient state can reach at least one absorbing state, ensuring eventual absorption with probability 1 from any starting state.^[12]^[2]^[14] The structure is derived by permuting the rows and columns of the original transition matrix according to the specified state ordering, as detailed in the state rearrangement process.^[13]^[7]

Fundamental Matrix

Construction

The fundamental matrix N for an absorbing Markov chain is constructed from the canonical form of its transition matrix P, specifically using the submatrix Q that governs transitions among the transient states.^[2] This matrix is defined as

N = (I - Q)^{-1},

where I denotes the t \times t identity matrix and t is the number of transient states.^[2] The resulting N is a t \times t matrix whose entry n_{ij} gives the expected number of visits to transient state j before absorption, starting from transient state i.^[2] The matrix I - Q is invertible because, in an absorbing Markov chain where absorption occurs with probability 1 from every state, the powers Q^n converge to the zero matrix as n \to \infty, implying that the spectral radius of Q is strictly less than 1.^[2]^[4] For computation with a finite number t of transient states, N can be obtained by direct matrix inversion of I - Q; in cases of larger t, numerical techniques such as Gaussian elimination are employed to solve the system efficiently.^[2]^[4]

Basic Interpretation

The fundamental matrix N of an absorbing Markov chain provides a probabilistic interpretation of the expected behavior within the transient states prior to absorption. Specifically, the entry N_{ij} represents the expected number of visits to transient state j starting from transient state i, including the initial visit if i = j, before the chain reaches an absorbing state.^[5] This interpretation arises because N sums the geometric series of powers of the transient transition submatrix Q, capturing the cumulative visits over all possible steps until absorption occurs.^[5] Diagonal entries N_{ii} are generally greater than 1, reflecting the possibility of multiple returns to the starting state i before absorption, as the chain may loop through transient states.^[5] For instance, in a simple gambler's ruin model with absorbing barriers, the expected visits to the initial fortune level exceed one due to potential oscillations. The row sums \sum_j N_{ij} yield the expected total time spent in all transient states starting from state i, which equals the expected steps to absorption from i.^[5] Since absorption is certain in an absorbing Markov chain—all transient states lead inevitably to an absorbing state—the powers of Q converge to zero, ensuring that all entries of N are finite and positive.^[5] This finiteness underscores the transient nature of these states, distinguishing absorbing chains from recurrent ones where expected visits could be infinite.^[5]

Derived Quantities

Absorption Probabilities

In an absorbing Markov chain, absorption probabilities quantify the likelihood that the process, starting from a given transient state, will eventually reach a specific absorbing state. These probabilities are fundamental to analyzing the long-term behavior of the chain, as absorption is certain in such systems, ensuring that the process terminates in one of the absorbing states with probability 1. The absorption probability matrix B is defined such that its entry B_{ik} represents the probability of being absorbed in absorbing state k when starting from transient state i. This matrix is computed as B = NR, where N is the fundamental matrix and R is the submatrix of transition probabilities from transient states to absorbing states. A key property of B is that each row sums to 1, reflecting the fact that the process must be absorbed in exactly one absorbing state from any transient starting point. For initial states that are already absorbing, the absorption probabilities form the identity matrix: the probability is 1 for the starting absorbing state j and 0 for all others. Computation of B involves forming the fundamental matrix N = (I - Q)^{-1}, where Q is the transient-to-transparent transition submatrix, and then multiplying N by R. In the limit as the number of steps approaches infinity, the stationary distribution over the absorbing states coincides with these absorption probabilities, weighted by the initial distribution on transient states. When multiple absorbing states are present, the rows of B enable direct comparison of the relative likelihoods of different absorption endpoints, facilitating the ranking of outcomes based on their probabilities from each transient state.

Expected Steps to Absorption

In an absorbing Markov chain, the expected number of steps to absorption from a given transient state quantifies the average duration the process remains in the set of transient states before entering an absorbing state. This measure, denoted t_i for starting state i, is a fundamental quantity that arises naturally from the chain's transition structure and provides insight into the typical lifetime of transient behavior.^[2]^[15] To compute t_i, consider the canonical form of the transition matrix, where the submatrix Q governs transitions among the transient states. The expected steps satisfy the vector equation \mathbf{t} = \mathbf{1} + Q \mathbf{t}, where \mathbf{t} is the column vector of t_i values and \mathbf{1} is the column vector of ones with dimension equal to the number of transient states. This system of linear equations reflects the fact that from any transient state, one step is taken immediately, followed by an expected additional Q \mathbf{t} steps from the resulting state. Solving yields \mathbf{t} = (I - Q)^{-1} \mathbf{1}, where I is the identity matrix. Equivalently, if N = (I - Q)^{-1} is the fundamental matrix, then the i-th entry is t_i = \sum_j N_{ij}, the sum of the i-th row of N. These expressions, introduced in the seminal treatment of finite Markov chains, allow efficient computation even for larger state spaces.^[2]^[15] The interpretation of t_i emphasizes that it counts only the time spent in transient states, including the starting step; upon reaching an absorbing state, the expected remaining steps is zero, as the process halts there. The fundamental matrix N facilitates this by encoding the expected number of visits to each transient state before absorption, with the row sum aggregating these into the total expected duration.^[2]^[1] In discrete-time Markov chains, t_i is measured in the number of transitions, providing a dimensionless count of steps; analogous concepts in continuous-time chains involve expected sojourn times but are not addressed here.^[2]^[1]

Expected Visits to Transient States

In an absorbing Markov chain, the fundamental matrix N = (I - Q)^{-1}, where Q is the submatrix of transition probabilities among the transient states, provides the expected number of visits to each transient state before absorption. Specifically, the entry N_{ij} denotes the expected number of times the chain visits transient state j starting from transient state i, including the initial visit if i = j. This quantity captures the average occupancy of transient states along paths leading to absorption, offering insights into the chain's behavior prior to termination.^[2]^[1] For the diagonal entries, N_{ii} represents the expected number of visits to state i itself when starting from i, which always includes at least one visit at time zero and accounts for potential returns before absorption. Off-diagonal entries N_{ij} (for i \neq j) measure the expected visits to state j via paths originating from i, reflecting the influence of the chain's structure on transient trajectories. These pairwise expectations form the full matrix N, enabling computation of visit distributions across all transient states for any starting configuration.^[2] The row sums of N provide a key normalization: for each starting transient state i, \sum_j N_{ij} equals the expected total number of steps spent in transient states before absorption, often denoted t_i. This links the per-state visits to the overall duration of the transient phase, confirming that the matrix entries aggregate to the chain's expected lifetime in non-absorbing states.^[2] These expected visits find application in reliability analysis, where transient states model system degradations and N_{ij} predicts time spent in specific failure modes before repair (absorption), and in queueing systems, where they forecast occupancies in buffer states prior to service completion.

Variances in Steps and Visits

In absorbing Markov chains, the variance of the number of steps to absorption provides a measure of the dispersion around the expected absorption time. For a starting transient state i, the variance \operatorname{Var}(t_i) is the i-th entry of the vector \operatorname{Var}(\mathbf{t}) = (2N - I) \mathbf{t} - \mathbf{t} \circ \mathbf{t}, where N is the fundamental matrix, \mathbf{t} = N \mathbf{1} is the vector of expected steps (with \mathbf{1} the all-ones vector), I is the identity matrix, and \circ denotes elementwise multiplication.^[16] Similarly, the variance of the number of visits to a transient state j starting from transient state i is \operatorname{Var}(n_{ij}) = N_{ij}(2N_{jj} - 1) - N_{ij}^2, where N_{ij} is the expected number of visits to j from i. This is the (i,j)-entry of the matrix N(2N_{\mathrm{dg}} - I) - N \circ N, with N_{\mathrm{dg}} the diagonal matrix of N.^[16] These variances quantify the uncertainty in the duration of the transient phase and the occupancy of specific states before absorption; for instance, higher variances occur in chains featuring long or branching transient paths that allow for greater variability in trajectories.^[16] The derivations assume a finite number of states, ensuring the fundamental matrix N exists and is finite; in infinite-state absorbing chains, variances may be infinite if the expected times are infinite.^[16]

Applications

Games of Chance

Absorbing Markov chains provide a natural framework for analyzing probabilistic games of chance, where outcomes lead to terminal states. A canonical example is the gambler's ruin problem, in which a gambler begins with initial capital k (where $0 < k < N) and repeatedly bets against an adversary holding the remaining capital up to a total of N units. Each bet results in the gambler gaining 1 unit with probability p or losing 1 unit with probability q = 1 - p, modeling the states as capital levels \{0, 1, \dots, N\}, with 0 (ruin) and N (victory) as absorbing states; from transient states i = 1, \dots, N-1, the transition probabilities are P_{i,i+1} = p and P_{i,i-1} = q.^[17] In this model, the probability of absorption at the ruin state 0, starting from capital k, is r_k = \frac{(q/p)^k - (q/p)^N}{1 - (q/p)^N} when p \neq q, and r_k = (N - k)/N in the fair case where p = q = 1/2. These probabilities, which quantify the gambler's risk of bankruptcy versus reaching the goal, can be derived by solving the system of linear equations from the balance conditions at each transient state.^[17] The expected duration of the game, or number of steps until absorption starting from k, is E[T_k] = k(N - k) for the fair game (p = 1/2); for the biased case (p \neq 1/2), it takes the closed-form expression

E[T_k] = \frac{1}{q - p} \left( N \frac{1 - (q/p)^k}{1 - (q/p)^N} - k \right). $$ This measures the average length of play before resolution, highlighting how bias affects persistence in the game.[](https://sites.math.rutgers.edu/~zeilberg/EM20/GamblersRuin.pdf) The [gambler's ruin](/page/Gambler's_ruin) problem originated in the 1654 correspondence between [Blaise Pascal](/page/Blaise_Pascal) and [Pierre de Fermat](/page/Pierre_de_Fermat), who addressed related gambling dilemmas to establish early principles of probability, though its interpretation as an absorbing Markov chain emerged later with the development of [stochastic](/page/Stochastic) processes in the [20th century](/page/20th_century).[](https://www.jstor.org/stable/1402732) ### Infectious Disease Modeling Absorbing Markov chains provide a framework for modeling diagnostic testing processes in infectious diseases, where the goal is to detect [infection](/page/Infection) status amid uncertainties like false negatives. In such models, transient states represent undiagnosed individuals, categorized by true [infection](/page/Infection) status—susceptible or infected—while absorbing states correspond to test outcomes: tested positive (leading to [quarantine](/page/Quarantine)) or tested negative (leading to clearance). Transitions occur through [infection](/page/Infection) events or testing actions, with probabilities governed by [infection](/page/Infection) rates (e.g., contact-based [transmission](/page/Transmission)) and test accuracy metrics, including [sensitivity](/page/Sensitivity) (true positive rate) and specificity (true negative rate). For an infected individual, repeated testing allows the chain to persist in the transient "infected undiagnosed" state with probability equal to the false negative rate per test, until absorption in the positive state via a true positive result. The [absorption](/page/Absorption) probabilities in these models quantify the likelihood of eventual correct [diagnosis](/page/Diagnosis) given the true [status](/page/Status). For instance, starting from an infected undiagnosed [state](/page/State), the probability of [absorption](/page/Absorption) in the tested positive [state](/page/State) incorporates the false negative [rate](/page/Rate), yielding $ b = \text{[sensitivity](/page/Sensitivity)} $ in simplified single-test cases, but extends to repeated testing scenarios where cumulative detection approaches 1 as the number of tests increases, adjusted for test [sensitivity](/page/Sensitivity) (e.g., 70-90% for [COVID-19](/page/COVID-19) RT-PCR). This metric helps evaluate the reliability of testing protocols, highlighting how false negatives delay [isolation](/page/Isolation) and sustain [transmission](/page/Transmission). In susceptible individuals, [absorption](/page/Absorption) in the tested negative [state](/page/State) is near-certain, but false positives can lead to unnecessary [quarantine](/page/Quarantine), balanced by specificity rates above 95%. These probabilities are derived using the fundamental [matrix](/page/Matrix) of the absorbing [chain](/page/Chain), enabling computation of long-term outcomes without [simulation](/page/Simulation).[](https://pmc.ncbi.nlm.nih.gov/articles/PMC8793126/)[](https://www.nature.com/articles/s41598-021-89127-1) Expected time to absorption represents the anticipated duration until reliable diagnosis, interpreted as steps (e.g., testing intervals) to reach an absorbing state. In the infected case, this equals the expected number of tests needed to overcome false negatives, following a geometric distribution with success probability equal to test sensitivity, yielding values approximately $ 1 / \text{sensitivity} $, e.g., 1.1-1.4 tests for sensitivities around 80%. For population-level models, it informs quarantine durations or retesting schedules to minimize undiagnosed spread. This quantity is computed via the fundamental matrix $ N = (I - Q)^{-1} $, where the sum of row entries gives the expected visits (and thus time) in transient states before absorption. A prominent real-world application arose during the COVID-19 pandemic (circa 2020), where absorbing Markov chains modeled testing and quarantine dynamics in high-risk settings like universities. In one such framework for college reopening, states included arrival testing, isolation (for positives), and absorbing outcomes like quarantine clearance or program completion/expulsion, with transitions reflecting test accuracy and infection risks. Absorption in quarantine (positive test) captured isolation of detected cases, while clearance (negative after monitoring) absorbed negatives, incorporating false negative risks through retesting probabilities; expected time to absorption aligned with 7-14 day quarantine protocols to ensure reliable status confirmation. This approach predicted expulsion risks around 22% under baseline testing, aiding resource allocation for containment.[](https://higherlogicdownload.s3.amazonaws.com/AMSTAT/8756478b-683a-475b-9af7-bf93c9147a3a/UploadedImages/MarkovChainForCovid2020.pdf) ### Sequence Generation Processes Absorbing Markov chains provide a framework for modeling the generation of random sequences until a specific pattern is achieved, where states represent the progress toward matching the target pattern. In this setup, transient states correspond to partial matches of the desired string, and the absorbing state is reached upon completing the full pattern. For instance, consider generating a sequence of fair coin flips (heads H or tails T, each with probability 1/2) until the pattern "HH" appears. The states are defined as: state ∅ (no match or starting state), state H (the last flip was H but not preceded by another H), and state HH (absorbing state, full match). Transitions occur as follows: from ∅, a T returns to ∅ (probability 1/2), while an H moves to H (probability 1/2); from H, a T returns to ∅ (probability 1/2), and an H moves to HH (probability 1/2); from HH, it remains there with probability 1. The [transition matrix](/page/Transition_matrix) for the transient states (∅ and H) forms the submatrix [Q](/page/Q), from which the fundamental matrix [N](/page/N+) = (I - [Q](/page/Q))^{-1} is computed to determine the expected number of steps to [absorption](/page/Absorption). For the "HH" example, solving the system yields an expected length of 6 flips from the starting state ∅. This approach extends to longer or more complex patterns, such as "HTH," where states track overlapping partial matches (e.g., after HT, an H completes HTH but also restarts a potential H). The expected steps can be found similarly using [N](/page/N+), revealing values like 10 for "HTH" in [fair coin](/page/Fair_coin) flips. In applications, this methodology is used to compute waiting times for specific nucleotide patterns in DNA sequences, modeled as i.i.d. or Markov-dependent trials over the alphabet {A, C, G, T}. For example, absorbing chains help estimate the expected length of a genomic sequence until a restriction enzyme recognition site (a short motif like "GAATTC") appears, aiding in gene synthesis and sequence design. Similarly, in text generation or natural language processing, the framework analyzes the expected length until rare word patterns or phrases emerge in random or probabilistic text streams, informing models for autocorrect or predictive typing. Expected visits to transient states can briefly indicate the frequency of partial matches during generation.[](https://www.sciencedirect.com/science/article/abs/pii/S0167715210002270)

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