en>Melcombe: move characteristic functions as they don't define the process

2013-03-29T17:51:38Z

move characteristic functions as they don't define the process

New page

In [[probability theory]], '''Kelly's lemma''' states that for a stationary [[continuous time Markov chain]], a process defined as the time-reversed process has the same stationary distribution as the forward-time process.<ref name="bou">{{cite book | page = 222 | title = Queueing Networks: A Fundamental Approach | first1 = Richard J. |last1= Boucherie | first2= N. M. | last2 = van Dijk | publisher = Springer | year = 2011 | isbn = 144196472X}}</ref> The theorem is named after [[Frank Kelly (mathematician)|Frank Kelly]].<ref>{{cite book | page = 22 | title = Reversibility and Stochastic Networks | url = http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html | first = Frank P. | last = Kelly | authorlink = Frank P. Kelly | year = 1979 | publisher = J. Wiley | isbn = 0471276014}}</ref><ref>{{cite book | page = 63 (Lemma 2.8.5) | title = An introduction to queueing networks | first = Jean | last = Walrand | year = 1988 | publisher = Prentice Hall | isbn = 013474487X}}</ref><ref>{{cite jstor|1425912}}</ref><ref>{{cite doi|10.1007/0-387-21525-5_2}}</ref>

==Statement==

For a continuous time Markov chain with state space ''S'' and [[transition rate matrix]] ''Q'' (with elements ''q''''ij'') if we can find a set of numbers ''q'''''ij'' and ''π''''i'' summing to 1 where<ref name="bou" />
::<math>\begin{align}
\sum_{j \neq i} \pi_i q'_{ij} &= \sum_{j \neq i} q_{ij} \quad \forall i\in S\\
\pi_i q_{ij} &= \pi_jq_{ji}' \quad \forall i,j \in S
\end{align}</math>
then ''q'''''ij'' are the rates for the reversed process and ''π''''i'' are the stationary distribution for both processes.

===Proof===

Given the assumptions made on the ''q''''ij'' and ''π''''i'' we can see
::<math> \sum_{i \neq j} \pi_i q_{ij} = \sum_{i \neq j} \pi_j q'_{ji} = \pi_j \sum_{i \neq j} q_{ji} = -\pi_j q_{jj}</math>
so the [[global balance equation]]s are satisfied and the ''π''''i'' are a stationary distribution for both processes.

==References==

{{Reflist}}

[[Category:Stochastic processes]]
[[Category:Queueing theory]]

Cauchy process - Revision history

en>Melcombe: move characteristic functions as they don't define the process