Leslie matrix stationary distribution Given a particular normalized Perron vector representing a discrete probability distribution, is it possible to derive some constraints or particular Leslie matrices having the given as their Perron vector?
There is a related question on math overflow.
 A: I have very little knowledge about demography. Yet, if the Leslie matrices you talk about are the ones described in this Wikipedia page, it seems like that for any given $v=(v_0,v_1,\ldots,v_{\omega  - 1})^T$, a corresponding Leslie matrix exists if $v_0>0$ and $v_0\ge v_1\ge\ldots\ge v_{\omega  - 1}\ge0$.
For such a vector $v$, let $v_j$ be the smallest nonzero entry (i.e. $j$ is the largest index such that $v_j>0$). Define
$$
s_i = \begin{cases}\frac{v_{i+1}}{v_i}&\ \textrm{ if } v_i>0,\\
\textrm{any number } \in[0,1]&\ \textrm{ otherwise}.\end{cases}
$$
and let $f=(f_0,f_1,\ldots,f_{\omega-1})^T$ be any entrywise nonnegative vector such that
$$
f_0 + \sum_{i=1}^{\omega-1}s_0s_1\ldots s_{i-1}f_i = 1.
$$
Then the Euler-Lokta equation
$$
f_0 + \sum_{i=1}^{\omega-1}\frac{s_0s_1\ldots s_{i-1}f_i}{\lambda^i} = \lambda$$
is satisfied and hence $v$, up to a normalizing factor, is the stable age distribution or Perron eigenvector of the Leslie matrix
$$
L = \begin{bmatrix}
f_0 & f_1 & f_2 & f_3 & \ldots &f_{\omega  - 1} \\
s_0 & 0 & 0 & 0 & \ldots & 0\\
0 & s_1 & 0 & 0 & \ldots & 0\\
0 & 0 & s_2 & 0 & \ldots & 0\\
0 & 0 & 0 & \ddots & \ldots & 0\\
0 & 0 & 0 & \ldots & s_{\omega - 2}  & 0
\end{bmatrix}.
$$
