The concept of duality and applications to Markov processes arising in neutral population genetics models. (English) Zbl 0942.92020

Möhle, Martin

Bernoulli 5, No. 5, 761-777 (1999).

The paper considers neutral population genetic models with non-overlapping generations, one for haploid populations (with fixed population size \(N)\) and one for a two-sex population (with \(N\) pairs fixed size). Both the forward (number of descendents of a given set of individuals) and the backward processes are discrete-time homogeneous Markov chains with state space \(\{0,1, \dots, N\}\). The forward process is dual to the backward process with respect to certain \((N+1) \times(N+1)\) matrices \(H\), in the sense that \(\Pi H=HP'\), where \(\Pi\) and \(P\) are the one-step transition matrices of the forward and backward processes, respectively. This duality can be quite useful to deduce certain properties of the forward process (particularly those resulting from the spectral properties of \(\Pi)\) from properties of the backward process that might be easier to obtain. The dimension of the duality space \(U\) the space of matrices \(H\) for which the duality holds, is related to the spectral properties of \(\Pi\) and \(P\).
The paper obtains, for both models considered, certain results on that regard. It also obtains, under certain conditions, basis for the linear space \(U\). An application of these results relates stationary distributions for the backward processes with extinction probabilities for the forward process.

Reviewer: Carlos A.Braumann (Evora)

MSC:

92D10	Genetics and epigenetics
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60J05	Discrete-time Markov processes on general state spaces
60J20	Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

Keywords:

duality space; haploid population models; two-sex population models; neutral population genetic models