Probability, statistics

Let r_ij(n)=P(X_n = j | X_0=i) be the n-step transition probability of a given homogeneous discrete-time Markov chain with m states. We have shown that r_ij(n) satisfies the following recursion for n>=2:
r_ij(n) = sum_(k=1)^(m) r_ik(n-1)*p_kj
For each of the following, decide whether it is also a valid recursion formula for r_ij(n).

1. r_{ij}(n)=\sum _{k=1}^ m p_{ik}r_{kj}(n-1) for n>=2

2. r_{ij}(n)=\sum _{k=1}^ m r_{ik}(n-2)r_{kj}(2) for n>=3

3. r_{ij}(n)=\sum _{k=1}^ m \sum _{\ell =1}^ m r_{ik}(n-2)p_{k\ell }p_{\ell j} for n>=3

  1. 👍
  2. 👎
  3. 👁
  4. ℹ️
  5. 🚩
  1. yes,
    yes
    yes

    1. 👍
    2. 👎
    3. ℹ️
    4. 🚩

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