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Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one defines $d_n=a_n-a_{n+1}$ which then leads to$$d_n=\frac{\alpha\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}{\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}$$ where $\alpha$ is a constant, and $$a_n=1-\sum_{j=0}^{n-1}d_j$$ whenever the infinite sum converges (otherwise, $a_n=1$ is the correct solution). To fixestablish the value of $\alpha$, there is a 'simple probabilistic argument' (Karlin) that $\alpha$$\lim_{n\to\infty}a_n$ cannot be positive and less than 1; therefore, itthus must be equal to 1, implyingzero (implying that $\lim_{n\to\infty}a_n=0$$\alpha=1$). My question is: what is that 'simple probabilistic argument'argument?

Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one defines $d_n=a_n-a_{n+1}$ which then leads to$$d_n=\frac{\alpha\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}{\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}$$ and $$a_n=1-\sum_{j=0}^{n-1}d_j$$ whenever the infinite sum converges (otherwise, $a_n=1$ is the correct solution). To fix the value of $\alpha$, there is a 'simple probabilistic argument' (Karlin) that $\alpha$ cannot be positive and less than 1; therefore, it must equal to 1, implying that $\lim_{n\to\infty}a_n=0$. My question is: what is that 'simple probabilistic argument'?

Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one defines $d_n=a_n-a_{n+1}$ which then leads to$$d_n=\frac{\alpha\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}{\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}$$ where $\alpha$ is a constant, and $$a_n=1-\sum_{j=0}^{n-1}d_j$$ whenever the infinite sum converges (otherwise, $a_n=1$ is the correct solution). To establish the value of $\alpha$, there is a 'simple probabilistic argument' (Karlin) that $\lim_{n\to\infty}a_n$ cannot be positive, thus must be equal to zero (implying that $\alpha=1$). My question is: what is that argument?

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Honza
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Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one can show that the correspondingdefines $\lim_{n\to\infty}a_n$ is not allowed$d_n=a_n-a_{n+1}$ which then leads to be positive$$d_n=\frac{\alpha\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}{\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}$$ and $$a_n=1-\sum_{j=0}^{n-1}d_j$$ whenever the infinite sum converges (thusotherwise, can only be equal to 0$a_n=1$ is the correct solution) 'by. To fix the value of $\alpha$, there is a simple'simple probabilistic argument' (claims Karlin, without spelling it outKarlin). This then adds the second 'boundary' condition on the desired solution that $\alpha$ cannot be positive and less than 1; therefore, thus making it uniquemust equal to 1, implying that $\lim_{n\to\infty}a_n=0$. What exactly My question is: what is that 'simple probabilistic argument'?

Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one can show that the corresponding $\lim_{n\to\infty}a_n$ is not allowed to be positive (thus, can only be equal to 0) 'by a simple probabilistic argument' (claims Karlin, without spelling it out). This then adds the second 'boundary' condition on the desired solution, thus making it unique. What exactly is that 'simple probabilistic argument'?

Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a_n$) is computed by solving $$a_n=\frac{\lambda_na_{n+1}+\mu_na_{n-1}}{\lambda_n+\mu_n}$$ for $n\ge1$, where $\lambda_n$ and $\mu_n$ are the up and down rates (respectively), all positive, and $a_0=1$. Seeking a solution other than $a_n\equiv1$ one defines $d_n=a_n-a_{n+1}$ which then leads to$$d_n=\frac{\alpha\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}{\sum_{n=0}^\infty\prod_{j=1}^n\frac{\mu_j}{\lambda_j}}$$ and $$a_n=1-\sum_{j=0}^{n-1}d_j$$ whenever the infinite sum converges (otherwise, $a_n=1$ is the correct solution). To fix the value of $\alpha$, there is a 'simple probabilistic argument' (Karlin) that $\alpha$ cannot be positive and less than 1; therefore, it must equal to 1, implying that $\lim_{n\to\infty}a_n=0$. My question is: what is that 'simple probabilistic argument'?

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