Chapman-kolmogorov equation

In probability theory, especially in stochastic process theory, the Chapman-kolmogorov equation is an important conclusion. It links the joint distribution functions of several different dimensions of a stochastic process. If the stochastic process is specific to a Markov chain, then the Chapman-kolmogorov equation is the formula for the transfer probability. I hope you understand the following two preliminary articles before reading this article: the limit of matrix and the limit of Markov chain (upper) Matrix and Markov chain (bottom)

Let's start with an example, the following is an example of the matrix limit and Markov chain (top). On the left side of the graph is the state transition matrix, where each position represents the probability of moving from one state to another state. For example, the probability of moving from State 1 to state 2 is 0.28, so the value given is 0.28 in the position of the second column in the first row.

In the Markov chain, the random variable in a time-sorted array T 1, t 2,..., t N, according to the state transition matrix let us be very intuitive to know the current moment a state at the next moment to change the probability of any state, as shown in the figure below.

Now our question is whether we can make further predictions. For example, if the current time T1 state is a, can we know what the probability of the state being B at the next moment T3? This is actually quite easy to do, as shown in the figure below.

This process is very intuitive and easy to represent with the squared of the transfer matrix. The 1th line in the matrix p indicates the probability that the current moment state a changes to the state a,b,c at the next moment, and the 2nd column in the matrix p represents the probability that the state a,b,c, the next moment, to move to State B. Then, according to the multiplication formula of the matrix, the position of the 2nd column in the 1th row of P2 means "the T1 state at the current moment is a, and the probability that the state is B at the next moment T3". That is, if we want a transfer matrix spanning 2 moments, multiply the transfer matrix across 1 times by the transfer matrix spanning 1 moments. Similarly, if you want to span a 3-moment transfer matrix, multiply the transfer matrix across 2 times by multiplying the transfer matrix across 1 moments. More generally, we have pm+n= PMPN, and this relationship is called Chapman-kolmogorov equation.

Here is a description of Chapman-kolmogorov equation:

For Markov chains, it should be expressed in discrete form, and for more generalized stochastic processes, it should be expressed in a continuous form:

Here is the proof of Chapman-kolmogorov equation.

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