Abstract
| Original language | Undefined |
|---|---|
| Article number | 10.1016/j.laa.2008.04.004 |
| Pages (from-to) | 776-791 |
| Number of pages | 14 |
| Journal | Linear algebra and its applications |
| Volume | 429 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2008 |
Keywords
- EWI-12808
- MSC-60J27
- M-matrix
- Quasi-stationary distribution
- survival-time distribution
- Absorbing Markov chain
- death process
- migration process
- METIS-250995
- IR-62322
- limiting conditional distribution
Cite this
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Survival in a quasi-death process. / van Doorn, Erik A.; Pollett, Philip K.
In: Linear algebra and its applications, Vol. 429, No. 4, 10.1016/j.laa.2008.04.004, 2008, p. 776-791.Research output: Contribution to journal › Article › Academic › peer-review
TY - JOUR
T1 - Survival in a quasi-death process
AU - van Doorn, Erik A.
AU - Pollett, Philip K.
N1 - 10.1016/j.laa.2008.04.004
PY - 2008
Y1 - 2008
N2 - We consider a Markov chain in continuous time with an absorbing coffin state and a finite set $S$ of transient states. When $S$ is irreducible the limiting distribution of the chain as $t \to\infty,$ conditional on survival up to time $t,$ is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which $S$ may be reducible, and show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of $S$ satisfy an accessibility constraint. A key role in the analysis is played by some classic results on $M$-matrices.
AB - We consider a Markov chain in continuous time with an absorbing coffin state and a finite set $S$ of transient states. When $S$ is irreducible the limiting distribution of the chain as $t \to\infty,$ conditional on survival up to time $t,$ is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which $S$ may be reducible, and show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on $S$ has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of $S$ satisfy an accessibility constraint. A key role in the analysis is played by some classic results on $M$-matrices.
KW - EWI-12808
KW - MSC-60J27
KW - M-matrix
KW - Quasi-stationary distribution
KW - survival-time distribution
KW - Absorbing Markov chain
KW - death process
KW - migration process
KW - METIS-250995
KW - IR-62322
KW - limiting conditional distribution
U2 - 10.1016/j.laa.2008.04.004
DO - 10.1016/j.laa.2008.04.004
M3 - Article
VL - 429
SP - 776
EP - 791
JO - Linear algebra and its applications
JF - Linear algebra and its applications
SN - 0024-3795
IS - 4
M1 - 10.1016/j.laa.2008.04.004
ER -