Timeline for Computing probability of ultimate absorption in B&D processes
Current License: CC BY-SA 4.0
15 events
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| Feb 3, 2021 at 20:50 | vote | accept | Honza | ||
| Oct 30, 2019 at 2:19 | comment | added | Honza | @ Iosif Pinelis: I assume that Karlin had something more 'basic' in mind - your arguments are quite advanced for the level of his book - but he may have been 'bluffing' too; it's very unusual for him to leave a gap in his proof for the reader to supply - as far as I can tell, he did it only twice! | |
| Oct 30, 2019 at 1:47 | comment | added | Iosif Pinelis | @Honza : I don't know what argument Karlin had in mind. To me, the above argument came easily, but only after I saw the way to use the continuity of probability theorem -- which I did not see right away. | |
| Oct 30, 2019 at 1:40 | comment | added | Honza | @ Iosif Pinelis: A neat, complete and very rigorous derivation of $a_n$. Karlin was a lot more cavalier with his proof; that's why his is rather different form yours, and why I still have to wonder what was his 'simple argument'. | |
| Oct 29, 2019 at 6:19 | comment | added | Iosif Pinelis | @Honza : The "simple probabilistic argument" turns out to consist in the use of the continuity of probability theorem, as now done in the answer. | |
| Oct 29, 2019 at 6:14 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Oct 28, 2019 at 16:41 | comment | added | Honza | The exact reference is: Samuel Karlin "A first course in stochastic processes" Academic Press 1969, page 203. I'll edit my question to closely follow his arguments. | |
| Oct 28, 2019 at 3:18 | comment | added | Iosif Pinelis | @Honza : In general, $a_n\not\to0$. | |
| Oct 28, 2019 at 3:17 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Oct 28, 2019 at 2:58 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Oct 27, 2019 at 22:15 | comment | added | Honza | Yes, again, you are right, and that's how the actual solution is built (first solving for the sequence of the $a_{n-1}-a_n$ differences, which is now quite easy), but I don't see how this imposes any constraint on the rates. I'll include full reference to Karlin tomorrow. | |
| Oct 27, 2019 at 20:41 | comment | added | Iosif Pinelis | @Honza : The problem with "additional condition on the solution" is that, if you impose such a condition on the sequence $(a_n)$, you'll be thus imposing an additional condition on the rates $\lambda_n$ and $\mu_n$, because equation (1) implies that $\lambda_n/\mu_n=(a_{n-1}-a_n)/(a_n-a_{n+1})$ for all natural $n$. The latter imposition, on the rates, does not look good, though. So, again, it could help if you completely reproduced here all the relevant definitions and exact statements from that book or, at least, gave us an accessible reference to the book. | |
| Oct 27, 2019 at 20:34 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Oct 27, 2019 at 20:13 | comment | added | Honza | Thanks, right you are, it seems the zero limit is not a consequence, but an additional condition on the solution (whose necessity follows from a 'simple probabilistic argument') - I'll need to rephrase my question accordingly. | |
| Oct 27, 2019 at 19:39 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |