Chapman-Kolmogorov Equation

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Explanation of Chapman-Kolmogorov Equation

Let {X_n,n ≥ 0} be a homogeneous Markov Chain with Transition Probability Matrix P. 

the n-step TPM is P(n) = P_ij(n), where

P_ij(n) = P(X_n = j / X₀ = i) and P_ij(1) = P


Let 'm' and 'n' be non-negative integers. Then the property that holds are:

• P(m+n) = P(m).P(n)
• P(n) = Pⁿ  (The n-step TPM is equal to the nth power of one step TPM)

The (i,j)th entry of matrix P(m+n) is 

P_ij(m+n) = P(X_m+n = j / X₀ = i)    

Continuing on solving this equation using matrix property, we get

where I = State Space.
This is the Chapman-Kolmogorov Equation.

• This shows that In first  'm' steps, Markov Chain proceeds from state i to some intermediate state k , and in remaining n steps, it proceeds from state k to state j.
• It also shows that the Probability of steps from state k to state j does not depend upon the manner in which k was reached.
• Using the Chapman-Kolmogorov Equation, future higher Transition Probability Matrix can be found using the present lower Transition Probability Matrix.

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