generating function of numbers of visits up to time n
I want to determine the generating function of the numbers of visits up to a specific time. Formally: let ${X_{n}}_{n geq 0}$ be a markov chain on the state space $S = {1,ldots,r}$ with transition matrix $P = (p_{i,j})_{1 leq i,j leq r}$ and initial probability vector $alpha = (alpha_{i} : i in S)$ with $alpha_{i} = mathbb{P}(X_{0} = i)$. Let $q(n_{1},ldots,n_{r})$ be the probability that up to time $n-1$ the state $1$ is visited $n_{1}$-times,$ldots$,state $r$ is visited $n_{r}$-times and $F_{n}(x_{1},ldots,x_{r}) = sum_{n_{1}+ cdots +n_{r} = n}q(n_{1},ldots,n_{r})x_{1}^{n_{1}}cdots x_{r}^{n_{r}}$ be the corresponding generating function of $q(n_{1},ldots,n_{r})$. Now I want to show that
$$
F_{n}(x_{1},ldots,x_{r}) = (alpha_{1}x_{1},ldots,alpha_{r}x_{r}) left( P D_{r}right)^{n-1} e,
$$
where $e = (1,ldots,1)^{intercal}$ and $D$ is given by
$$
D =
begin{bmatrix}
x_{1} && \
& ddots &\
& & x_{r}
end{bmatrix}.
$$
Here is my solution:
$$
begin{align*}
sum_{n_{1} + cdots + n_{r} = n}q(n_{1},ldots,n_{r}) &= sum_{j in I}mathbb{P}(X_{n-1} = j) \
&= sum_{j in I}sum_{i in I}alpha_{i}p_{i,j}^{n-1} \
&= left(sum_{i in I}alpha_{i}p_{i,1}^{n-1},ldots,sum_{i in I}alpha_{i}p_{i,r}^{n-1} right) e \
&= alpha P^{(n-1)} e.
end{align*}
$$
Does this suffice as a proof and is it correct? Thanks for any help!
probability probability-theory proof-verification markov-chains generating-functions
edited 17 hours ago
Scientifica
6,37641335
asked 17 hours ago
wayne
492113
add a comment |
I want to determine the generating function of the numbers of visits up to a specific time. Formally: let ${X_{n}}_{n geq 0}$ be a markov chain on the state space $S = {1,ldots,r}$ with transition matrix $P = (p_{i,j})_{1 leq i,j leq r}$ and initial probability vector $alpha = (alpha_{i} : i in S)$ with $alpha_{i} = mathbb{P}(X_{0} = i)$. Let $q(n_{1},ldots,n_{r})$ be the probability that up to time $n-1$ the state $1$ is visited $n_{1}$-times,$ldots$,state $r$ is visited $n_{r}$-times and $F_{n}(x_{1},ldots,x_{r}) = sum_{n_{1}+ cdots +n_{r} = n}q(n_{1},ldots,n_{r})x_{1}^{n_{1}}cdots x_{r}^{n_{r}}$ be the corresponding generating function of $q(n_{1},ldots,n_{r})$. Now I want to show that
$$
F_{n}(x_{1},ldots,x_{r}) = (alpha_{1}x_{1},ldots,alpha_{r}x_{r}) left( P D_{r}right)^{n-1} e,
$$
where $e = (1,ldots,1)^{intercal}$ and $D$ is given by
$$
D =
begin{bmatrix}
x_{1} && \
& ddots &\
& & x_{r}
end{bmatrix}.
$$
Here is my solution:
$$
begin{align*}
sum_{n_{1} + cdots + n_{r} = n}q(n_{1},ldots,n_{r}) &= sum_{j in I}mathbb{P}(X_{n-1} = j) \
&= sum_{j in I}sum_{i in I}alpha_{i}p_{i,j}^{n-1} \
&= left(sum_{i in I}alpha_{i}p_{i,1}^{n-1},ldots,sum_{i in I}alpha_{i}p_{i,r}^{n-1} right) e \
&= alpha P^{(n-1)} e.
end{align*}
$$
Does this suffice as a proof and is it correct? Thanks for any help!
probability probability-theory proof-verification markov-chains generating-functions
edited 17 hours ago
Scientifica
6,37641335
asked 17 hours ago
wayne
492113
add a comment |
I want to determine the generating function of the numbers of visits up to a specific time. Formally: let ${X_{n}}_{n geq 0}$ be a markov chain on the state space $S = {1,ldots,r}$ with transition matrix $P = (p_{i,j})_{1 leq i,j leq r}$ and initial probability vector $alpha = (alpha_{i} : i in S)$ with $alpha_{i} = mathbb{P}(X_{0} = i)$. Let $q(n_{1},ldots,n_{r})$ be the probability that up to time $n-1$ the state $1$ is visited $n_{1}$-times,$ldots$,state $r$ is visited $n_{r}$-times and $F_{n}(x_{1},ldots,x_{r}) = sum_{n_{1}+ cdots +n_{r} = n}q(n_{1},ldots,n_{r})x_{1}^{n_{1}}cdots x_{r}^{n_{r}}$ be the corresponding generating function of $q(n_{1},ldots,n_{r})$. Now I want to show that
$$
F_{n}(x_{1},ldots,x_{r}) = (alpha_{1}x_{1},ldots,alpha_{r}x_{r}) left( P D_{r}right)^{n-1} e,
$$
where $e = (1,ldots,1)^{intercal}$ and $D$ is given by
$$
D =
begin{bmatrix}
x_{1} && \
& ddots &\
& & x_{r}
end{bmatrix}.
$$
Here is my solution:
$$
begin{align*}
sum_{n_{1} + cdots + n_{r} = n}q(n_{1},ldots,n_{r}) &= sum_{j in I}mathbb{P}(X_{n-1} = j) \
&= sum_{j in I}sum_{i in I}alpha_{i}p_{i,j}^{n-1} \
&= left(sum_{i in I}alpha_{i}p_{i,1}^{n-1},ldots,sum_{i in I}alpha_{i}p_{i,r}^{n-1} right) e \
&= alpha P^{(n-1)} e.
end{align*}
$$
Does this suffice as a proof and is it correct? Thanks for any help!
probability probability-theory proof-verification markov-chains generating-functions
edited 17 hours ago
Scientifica
6,37641335
asked 17 hours ago
wayne
492113
I want to determine the generating function of the numbers of visits up to a specific time. Formally: let ${X_{n}}_{n geq 0}$ be a markov chain on the state space $S = {1,ldots,r}$ with transition matrix $P = (p_{i,j})_{1 leq i,j leq r}$ and initial probability vector $alpha = (alpha_{i} : i in S)$ with $alpha_{i} = mathbb{P}(X_{0} = i)$. Let $q(n_{1},ldots,n_{r})$ be the probability that up to time $n-1$ the state $1$ is visited $n_{1}$-times,$ldots$,state $r$ is visited $n_{r}$-times and $F_{n}(x_{1},ldots,x_{r}) = sum_{n_{1}+ cdots +n_{r} = n}q(n_{1},ldots,n_{r})x_{1}^{n_{1}}cdots x_{r}^{n_{r}}$ be the corresponding generating function of $q(n_{1},ldots,n_{r})$. Now I want to show that
$$
F_{n}(x_{1},ldots,x_{r}) = (alpha_{1}x_{1},ldots,alpha_{r}x_{r}) left( P D_{r}right)^{n-1} e,
$$
where $e = (1,ldots,1)^{intercal}$ and $D$ is given by
$$
D =
begin{bmatrix}
x_{1} && \
& ddots &\
& & x_{r}
end{bmatrix}.
$$
Here is my solution:
$$
begin{align*}
sum_{n_{1} + cdots + n_{r} = n}q(n_{1},ldots,n_{r}) &= sum_{j in I}mathbb{P}(X_{n-1} = j) \
&= sum_{j in I}sum_{i in I}alpha_{i}p_{i,j}^{n-1} \
&= left(sum_{i in I}alpha_{i}p_{i,1}^{n-1},ldots,sum_{i in I}alpha_{i}p_{i,r}^{n-1} right) e \
&= alpha P^{(n-1)} e.
end{align*}
$$
Does this suffice as a proof and is it correct? Thanks for any help!
probability probability-theory proof-verification markov-chains generating-functions
probability probability-theory proof-verification markov-chains generating-functions
edited 17 hours ago
Scientifica
6,37641335
asked 17 hours ago
wayne
492113
edited 17 hours ago
Scientifica
6,37641335
asked 17 hours ago
wayne
492113
edited 17 hours ago
Scientifica
6,37641335
edited 17 hours ago
Scientifica
6,37641335
edited 17 hours ago
Scientifica
6,37641335
6,37641335
asked 17 hours ago
wayne
492113
asked 17 hours ago
wayne
492113
asked 17 hours ago
wayne
492113
492113
add a comment |
add a comment |
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