IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
For a time-homogeneous Markov chain, which statement about the n-step transition matrix P(n) is correct?
The n-step transition matrix is the one-step matrix raised to the nth power using matrix multiplication. This follows from the Chapman-Kolmogorov equations. Element-wise powers or multiples of P would not give valid probabilities with row sums of one.
- AP(n) equals n times P
- BP(n) equals the matrix P raised to the power nCorrect
- CP(n) equals the element-wise nth power of P
- DP(n) always has identical rows for every n
- P(n) has row sums equal to n
Explanation
Chapman-Kolmogorov gives P(m+n)=P(m)P(n), so P(n)=P^n by matrix multiplication. Element-wise powers do not preserve row sums of one. Rows are identical only in the limit for ergodic chains.
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