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61-936: Let { X n } {\displaystyle \{X_{n}\}} be a Markov chain on a general state space Ω {\displaystyle \Omega } with stochastic kernel K {\displaystyle K} . The kernel represents a generalized one-step transition probability law, so that P ( X n + 1 ∈ C ∣ X n = x ) = K ( x , C ) {\displaystyle P(X_{n+1}\in C\mid X_{n}=x)=K(x,C)} for all states x {\displaystyle x} in Ω {\displaystyle \Omega } and all measurable sets C ⊆ Ω {\displaystyle C\subseteq \Omega } . The chain { X n } {\displaystyle \{X_{n}\}}

122-403: A countable set S called the state space of the chain. A continuous-time Markov chain ( X t ) t  ≥ 0 is defined by a finite or countable state space S , a transition rate matrix Q with dimensions equal to that of the state space and initial probability distribution defined on the state space. For i  ≠  j , the elements q ij are non-negative and describe

183-446: A kernel that is absolutely continuous with respect to Lebesgue measure : such that K ( x , y ) is a continuous function . Pick ( x 0 ,  y 0 ) such that K ( x 0 ,  y 0 ) > 0, and let A and Ω be open sets containing x 0 and y 0 respectively that are sufficiently small so that K ( x ,  y ) ≥ ε > 0 on A  ×  Ω. Letting ρ ( C ) = |Ω ∩  C |/|Ω| where |Ω|

244-443: A detailed study on Markov chains. Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener 's work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived

305-560: A diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé . Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938

366-406: A lengthy task. However, there are many techniques that can assist in finding this limit. Let P be an n × n matrix, and define Q = lim k → ∞ P k . {\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.} It is always true that Subtracting Q from both sides and factoring then yields where I n

427-409: A probability mass function on the states, so that ρ ( x ) ≥ 0 for all x ∈ Ω, and the sum of the ρ (x) probabilities is equal to one. Suppose the above definition is satisfied for a given set A ⊆ Ω and a given parameter ε > 0. Then P[ X n +1 = c | X n = x ] ≥ ερ ( c ) for all x ∈ A and all c ∈ Ω. Let { X n },  X n ∈ R be a Markov chain with

488-517: A quarter are drawn. Thus X 6 = $ 0.50 {\displaystyle X_{6}=\$ 0.50} . If we know not just X 6 {\displaystyle X_{6}} , but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel; so we can determine that X 7 ≥ $ 0.60 {\displaystyle X_{7}\geq \$ 0.60} with probability 1. But if we do not know

549-600: A rank-one matrix in which each row is the stationary distribution π : where 1 is the column vector with all entries equal to 1. This is stated by the Perron–Frobenius theorem . If, by whatever means, lim k → ∞ P k {\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}} is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below. For some stochastic matrices P ,

610-623: A set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation , in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to

671-424: Is a Harris chain if there exists A ⊆ Ω , ε > 0 {\displaystyle A\subseteq \Omega ,\varepsilon >0} , and probability measure ρ {\displaystyle \rho } with ρ ( Ω ) = 1 {\displaystyle \rho (\Omega )=1} such that The first part of the definition ensures that

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732-413: Is a left eigenvector of P and let Σ be the diagonal matrix of left eigenvalues of P , that is, Σ = diag( λ 1 , λ 2 , λ 3 ,..., λ n ). Then by eigendecomposition Let the eigenvalues be enumerated such that: Since P is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector

793-473: Is diagonalizable or equivalently that P has n linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is, defective matrices , one may start with the Jordan normal form of P and proceed with a bit more involved set of arguments in a similar way. ) Let U be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column

854-429: Is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution. The values of a stationary distribution π i {\displaystyle \textstyle \pi _{i}} are associated with

915-575: Is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC) , but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention. In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see Markov model ). Moreover,

976-418: Is not possible. After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state (since probabilistically important information has since been added to the scenario). In this way, the likelihood of the X n = i , j , k {\displaystyle X_{n}=i,j,k} state depends exclusively on

1037-401: Is one method for doing so: first, define the function f ( A ) to return the matrix A with its right-most column replaced with all 1's. If [ f ( P − I n )] exists then One thing to notice is that if P has an element P i , i on its main diagonal that is equal to 1 and the i th row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of

1098-483: Is possible to model this scenario as a Markov process. Instead of defining X n {\displaystyle X_{n}} to represent the total value of the coins on the table, we could define X n {\displaystyle X_{n}} to represent the count of the various coin types on the table. For instance, X 6 = 1 , 0 , 5 {\displaystyle X_{6}=1,0,5} could be defined to represent

1159-499: Is related to a Markov process. A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as " memorylessness "). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. In other words, conditional on

1220-474: Is the Kronecker delta , using the little-o notation . The q i j {\displaystyle q_{ij}} can be seen as measuring how quickly the transition from i to j happens. Define a discrete-time Markov chain Y n to describe the n th jump of the process and variables S 1 , S 2 , S 3 , ... to describe holding times in each of the states where S i follows

1281-501: Is the Lebesgue measure of Ω, we have that (2) in the above definition holds. If (1) holds, then { X n } is a Harris chain. In the following R := inf { n ≥ 1 : X n ∈ A } {\displaystyle R:=\inf\{n\geq 1:X_{n}\in A\}} ; i.e. R {\displaystyle R} is the first time after time 0 that

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1342-404: Is the identity matrix of size n , and 0 n , n is the zero matrix of size n × n . Multiplying together stochastic matrices always yields another stochastic matrix, so Q must be a stochastic matrix (see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that Q is a stochastic matrix to solve for Q . Including the fact that the sum of each

1403-414: Is the identity matrix . If the state space is finite , the transition probability distribution can be represented by a matrix , called the transition matrix, with the ( i , j )th element of P equal to Since each row of P sums to one and all elements are non-negative, P is a right stochastic matrix . A stationary distribution π is a (row) vector, whose entries are non-negative and sum to 1,

1464-430: Is unchanged by the operation of transition matrix P on it and so is defined by By comparing this definition with that of an eigenvector we see that the two concepts are related and that is a normalized ( ∑ i π i = 1 {\textstyle \sum _{i}\pi _{i}=1} ) multiple of a left eigenvector e of the transition matrix P with an eigenvalue of 1. If there

1525-533: Is unique too (because there is no other π which solves the stationary distribution equation above). Let u i be the i -th column of U matrix, that is, u i is the left eigenvector of P corresponding to λ i . Also let x be a length n row vector that represents a valid probability distribution; since the eigenvectors u i span R n , {\displaystyle \mathbb {R} ^{n},} we can write If we multiply x with P from right and continue this operation with

1586-405: Is unity and that π lies on a simplex . If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k -step transition probability can be computed as the k -th power of the transition matrix, P . If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π . Additionally, in this case P converges to

1647-424: The exponential distribution with rate parameter − q Y i Y i . For any value n = 0, 1, 2, 3, ... and times indexed up to this value of n : t 0 , t 1 , t 2 , ... and all states recorded at these times i 0 , i 1 , i 2 , i 3 , ... it holds that where p ij is the solution of the forward equation (a first-order differential equation ) with initial condition P(0)

1708-439: The integers or natural numbers , and the random process is a mapping of these to states. The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict with certainty

1769-1595: The Harris chain is called recurrent. Definition: A recurrent Harris chain X n {\displaystyle X_{n}} is aperiodic if ∃ N {\displaystyle \exists N} , such that ∀ n ≥ N {\displaystyle \forall n\geq N} , ∀ μ , P [ X n ∈ A | X 0 ∈ A ] > 0 {\displaystyle \forall \mu ,P[X_{n}\in A|X_{0}\in A]>;0} Theorem: Let X n {\displaystyle X_{n}} be an aperiodic recurrent Harris chain with stationary distribution π {\displaystyle \pi } . If P [ R < ∞ | X 0 ∈ A ] = 1 {\displaystyle P[R<\infty |X_{0}\in A]=1} then as n → ∞ {\displaystyle n\rightarrow \infty } , | | L ( X n | X 0 ) − π | | T V → 0 {\displaystyle ||{\mathcal {L}}(X_{n}|X_{0})-\pi ||_{TV}\rightarrow 0} where | | ⋅ | | T V {\displaystyle ||\cdot ||_{TV}} denotes

1830-610: The Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in Eugene Onegin , written by Alexander Pushkin , and proved a central limit theorem for such chains. In 1912 Henri Poincaré studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include

1891-506: The basis for general stochastic simulation methods known as Markov chain Monte Carlo , which are used for simulating sampling from complex probability distributions , and have found application in areas including Bayesian statistics , biology , chemistry , economics , finance , information theory , physics , signal processing , and speech processing . The adjectives Markovian and Markov are used to describe something that

Harris chain - Misplaced Pages Continue

1952-399: The chain returns to some state within A {\displaystyle A} with probability 1, regardless of where it starts. It follows that it visits state A {\displaystyle A} infinitely often (with probability 1). The second part implies that once the Markov chain is in state A {\displaystyle A} , its next-state can be generated with

2013-403: The definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are

2074-460: The earlier values, then based only on the value X 6 {\displaystyle X_{6}} we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses about X 7 {\displaystyle X_{7}} are impacted by our knowledge of values prior to X 6 {\displaystyle X_{6}} . However, it

2135-456: The early 20th century in the form of the Poisson process . Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with Pavel Nekrasov who claimed independence was necessary for the weak law of large numbers to hold. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of

2196-416: The first draw results in state X 1 = 0 , 1 , 0 {\displaystyle X_{1}=0,1,0} . The probability of achieving X 2 {\displaystyle X_{2}} now depends on X 1 {\displaystyle X_{1}} ; for example, the state X 2 = 1 , 0 , 1 {\displaystyle X_{2}=1,0,1}

2257-436: The foundations of Markov processes include William Feller , starting in 1930s, and then later Eugene Dynkin , starting in the 1950s. Suppose that there is a coin purse containing five quarters (each worth 25¢), five dimes (each worth 10¢), and five nickels (each worth 5¢), and one by one, coins are randomly drawn from the purse and are set on a table. If X n {\displaystyle X_{n}} represents

2318-542: The help of an independent Bernoulli coin flip. To see this, first note that the parameter ε {\displaystyle \varepsilon } must be between 0 and 1 (this can be shown by applying the second part of the definition to the set C = Ω {\displaystyle C=\Omega } ). Now let x {\displaystyle x} be a point in A {\displaystyle A} and suppose X n = x {\displaystyle X_{n}=x} . To choose

2379-408: The limit lim k → ∞ P k {\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}} does not exist while the stationary distribution does, as shown by this example: (This example illustrates a periodic Markov chain.) Because there are a number of different special cases to consider, the process of finding this limit if it exists can be

2440-448: The nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time: Note that there

2501-868: The next state X n + 1 {\displaystyle X_{n+1}} according to the measure P ( X n + 1 ∈ C ∣ X n = x ) = ( K ( x , C ) − ε ρ ( C ) ) / ( 1 − ε ) {\displaystyle P(X_{n+1}\in C\mid X_{n}=x)=(K(x,C)-\varepsilon \rho (C))/(1-\varepsilon )} (defined for all measurable subsets C ⊆ Ω {\displaystyle C\subseteq \Omega } ). Two random processes { X n } {\displaystyle \{X_{n}\}} and { Y n } {\displaystyle \{Y_{n}\}} that have

Harris chain - Misplaced Pages Continue

2562-581: The next state X n + 1 {\displaystyle X_{n+1}} , independently flip a biased coin with success probability ε {\displaystyle \varepsilon } . If the coin flip is successful, choose the next state X n + 1 ∈ Ω {\displaystyle X_{n+1}\in \Omega } according to the probability measure ρ {\displaystyle \rho } . Else (and if ε < 1 {\displaystyle \varepsilon <1} ), choose

2623-448: The outcome of the X n − 1 = ℓ , m , p {\displaystyle X_{n-1}=\ell ,m,p} state. A discrete-time Markov chain is a sequence of random variables X 1 , X 2 , X 3 , ... with the Markov property , namely that the probability of moving to the next state depends only on the present state and not on the previous states: The possible values of X i form

2684-429: The present state of the system, its future and past states are independent . A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of

2745-548: The process enters region A {\displaystyle A} . Let μ {\displaystyle \mu } denote the initial distribution of the Markov chain, i.e. X 0 ∼ μ {\displaystyle X_{0}\sim \mu } . Definition: If for all μ {\displaystyle \mu } , P [ R < ∞ | X 0 ∈ A ] = 1 {\displaystyle P[R<\infty |X_{0}\in A]=1} , then

2806-429: The rate of the process transitions from state i to state j . The elements q ii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one. There are three equivalent definitions of the process. Let X t {\displaystyle X_{t}} be the random variable describing

2867-632: The results, in the end we get the stationary distribution π . In other words, π = a 1 u 1 ← xPP ... P = xP as k → ∞. That means Since π is parallel to u 1 (normalized by L2 norm) and π is a probability vector, π approaches to a 1 u 1 = π as k → ∞ with a speed in the order of λ 2 / λ 1 exponentially. This follows because | λ 2 | ≥ ⋯ ≥ | λ n | , {\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,} hence λ 2 / λ 1

2928-415: The rows in P is 1, there are n+1 equations for determining n unknowns, so it is computationally easier if on the one hand one selects one row in Q and substitutes each of its elements by one, and on the other one substitutes the corresponding element (the one in the same column) in the vector 0 , and next left-multiplies this latter vector by the inverse of transformed former matrix to find Q . Here

2989-400: The same coin flip to decide the next-state of both processes, it follows that the next states are the same with probability at least ε {\displaystyle \varepsilon } . Let Ω be a countable state space. The kernel K is defined by the one-step conditional transition probabilities P[ X n +1 = y | X n  =  x ] for x , y ∈ Ω. The measure ρ is

3050-444: The same probability law and are Harris chains according to the above definition can be coupled as follows: Suppose that X n = x {\displaystyle X_{n}=x} and Y n = y {\displaystyle Y_{n}=y} , where x {\displaystyle x} and y {\displaystyle y} are points in A {\displaystyle A} . Using

3111-411: The state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important. Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906. Markov Processes in continuous time were discovered long before his work in

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3172-480: The state of affairs now ." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov . Markov chains have many applications as statistical models of real-world processes. They provide

3233-841: The state of the process at time t , and assume the process is in a state i at time t . Then, knowing X t = i {\displaystyle X_{t}=i} , X t + h = j {\displaystyle X_{t+h}=j} is independent of previous values ( X s : s < t ) {\displaystyle \left(X_{s}:s<t\right)} , and as h → 0 for all j and for all t , Pr ( X ( t + h ) = j ∣ X ( t ) = i ) = δ i j + q i j h + o ( h ) , {\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),} where δ i j {\displaystyle \delta _{ij}}

3294-417: The state space of P and its eigenvectors have their relative proportions preserved. Since the components of π are positive and the constraint that their sum is unity can be rewritten as ∑ i 1 ⋅ π i = 1 {\textstyle \sum _{i}1\cdot \pi _{i}=1} we see that the dot product of π with a vector whose components are all 1

3355-442: The state where there is one quarter, zero dimes, and five nickels on the table after 6 one-by-one draws. This new model could be represented by 6 × 6 × 6 = 216 {\displaystyle 6\times 6\times 6=216} possible states, where each state represents the number of coins of each type (from 0 to 5) that are on the table. (Not all of these states are reachable within 6 draws.) Suppose that

3416-452: The subsequent powers P . Hence, the i th row or column of Q will have the 1 and the 0's in the same positions as in P . As stated earlier, from the equation π = π P , {\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,} (if exists) the stationary (or steady state) distribution π is a left eigenvector of row stochastic matrix P . Then assuming that P

3477-406: The system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in

3538-497: The term may refer to a process on an arbitrary state space. However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see Variations ). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. The changes of state of

3599-436: The time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions:

3660-404: The total value of the coins set on the table after n draws, with X 0 = 0 {\displaystyle X_{0}=0} , then the sequence { X n : n ∈ N } {\displaystyle \{X_{n}:n\in \mathbb {N} \}} is not a Markov process. To see why this is the case, suppose that in the first six draws, all five nickels and

3721-465: The total variation for signed measures defined on the same measurable space. Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process 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

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