考慮兩個人口年齡層結構之簡單模型,且這兩個年齡層的產生率分別是m1及m2而且存活下來的機率是Pt.我們所要研究的是經過一大串隨機矩陣乘積運算後之結果

Consider a simple model of an age-structured population with two age-classes and stochastically varying survival rate of young. Let $m_{1,y},m_{2,t}$ be birth rates per capital and $P_{1,t}$ be a survival rate.

egin{eqnarray}
left(
egin{array}{clr}
N_{1,t+1}N_{2,t+1}
end{array}
ight)
=
left(
egin{array}{clr}
m_{1,t+1} & m_{2,t+1}
P_{1,t+1} & 0
end{array}
ight)
left(
egin{array}{clr}
N_{1,t}N_{2,t}
end{array}
ight)
end{eqnarray}

we want to study the large term behavior of $(N_{1,t},N_{2,t})$
the age-structured population through the theory of random matrix product.

Introduction
Random matrices
Simulation
Reference

H. Furstenberg and H. Kesten (1960), Products of Random Matrices, Ann. Math. Stat.,31,457-469.
J. F. C. Kingman (1968), The Ergodic Theory of Subadditive Stochastic Processes,J.Roy.Statist. Soc.Ser., B30, 499-510.
H. Furstenberg and Y. Kifer (1983), Random Matrix Products and Measures on Projective Spaces,Israel J.Math., 10, 12-32.
J. E. Cohen (1979), Ergodic Theorems in Demography, Amer.Math. Soc., Vol 1, 275-295.
Shripad Tuljapurkar(1986), Demographic Applications of Random Matrix Products,Contemporary Mathematics,Vol 50, 319-326.