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) = summation k = 1 to 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)= summation k=1 to m p_ik*r_kj (n-1) for n>=2
2. r_ij (n)=summation k=1 to m r_ik*(n-2)*r_kj (2) for n>=3
3. r_ij (n) = summation k=1 to m and l=1 to m r_ik*(n-2)*p_kl*p_lj for n>=3
1. This is a valid recursion formula for r_ij (n). It is similar to the given recursion formula, but with the transition probabilities and probabilities of being in state j and k swapped.
2. This is not a valid recursion formula for r_ij (n). The formula includes r_kj (2), which means it depends on the 2-step transition probabilities instead of the n-step transition probabilities.
3. This is a valid recursion formula for r_ij (n). It includes the 2-step transition probabilities r_ik (n-2) and r_kj (2), and also includes the transition probabilities p_kl and p_lj. It is a more complex version of the given recursion formula, but still satisfies the properties of a valid recursion formula.
1. Yes
2. Yes
3. Yes
I apologize for the incorrect response. Let's reassess each option again:
1. r_ij (n) = summation k=1 to m p_ik*r_kj (n-1) for n>=2: This is indeed a valid recursion formula for r_ij (n). It follows a similar structure to the given recursion formula and correctly calculates the n-step transition probability based on the (n-1)-step transition probabilities.
2. r_ij (n) = summation k=1 to m r_ik*(n-2)*r_kj (2) for n>=3: This is not a valid recursion formula for r_ij (n). The presence of r_kj (2) means it depends on the 2-step transition probabilities instead of the n-step transition probabilities, violating the necessary condition for a valid recursion formula.
3. r_ij (n) = summation k=1 to m and l=1 to m r_ik*(n-2)*p_kl*p_lj for n>=3: This is also not a valid recursion formula for r_ij (n). The double summation over k and l is not a valid approach for calculating the n-step transition probability. The formula is attempting to combine the (n-2)-step transition probabilities r_ik, transition probabilities p_kl, and p_lj, which deviates from the correct structure of a valid recursion formula.
Therefore, only option 1 is a valid recursion formula for r_ij (n).
Let's analyze each option to determine if it is a valid recursion formula for r_ij (n).
1. r_ij (n) = summation k=1 to m p_ik*r_kj (n-1) for n>=2
To verify if this formula is valid, we substitute the given recursion formula for r_kj (n-1) into it:
r_ij (n) = summation k=1 to m p_ik * (summation l=1 to m r_kl*(n-2) * p_lj) for n>=2
We can rewrite this summation as two separate summations:
r_ij (n) = summation k=1 to m summation l=1 to m (p_ik * r_kl*(n-2) * p_lj) for n>=2
This is equivalent to the given recursion formula:
r_ij (n) = summation k=1 to m and l=1 to m r_ik*(n-2)*p_kl*p_lj
Therefore, option 1 is a valid recursion formula for r_ij (n).
2. r_ij (n) = summation k=1 to m r_ik*(n-2)*r_kj (2) for n>=3
To determine if this formula is valid, we substitute the values for n=2 into it:
r_ij (2) = summation k=1 to m r_ik*(2-2)*r_kj (2)
This simplifies to:
r_ij (2) = summation k=1 to m r_ik * r_kj (2)
This expression is not the same as the given recursion formula, so option 2 is not a valid recursion formula for r_ij (n).
3. r_ij (n) = summation k=1 to m and l=1 to m r_ik*(n-2)*p_kl*p_lj for n>=3
This is the same as the given recursion formula, so option 3 is a valid recursion formula for r_ij (n).
In summary:
- Option 1 is a valid recursion formula.
- Option 2 is not a valid recursion formula.
- Option 3 is a valid recursion formula.