Gompertz distribution

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

In probability theory, Foster's theorem, named after F. G. Foster, is used to draw conclusions about the positive recurrence of Markov chains with countable state spaces. It uses the fact that positive recurrent Markov chains exhibit a notion of "Lyapunov stability" in terms of returning to any state while starting from it within a finite time interval.

Consider an aperiodic, irreducible discrete-time Markov chain on a countable state space $\mathbf{S}$ having a transition probability matrix $P=[p_{ij}]\mathrm{\forall }i,j\in \mathbf{S}$. Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function $V:\mathbf{S}\to \mathbf{R}$, such that

1. $V(i)\ge 0\mathrm{\forall }i\in \mathbf{S}$
2. $\sum _{j\in \mathbf{S}}{p}_{ij}V(j)\le V(i)-1$
3. $\sum _{j\in \mathbf{S}}{p}_{0j}V(j)<\mathrm{\infty }.$

Related links

• Lyapunov optimization
• Lyapunov function

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

1. Pierre Brémaud - Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer, 1998 (Chapter 5)