Let $\lambda_n$ denote the arrival rate for a pure birth process of size $n$. Let $P_n(t)$ denote the probability of population size $n$ at time $t$.
A stochastic process is dishonest if $\sum\limits_{j=0}^{\infty}P_j(t)<1$ for any $t$. If $\sum\limits_{j=0}^{\infty}P_j(t)=1$, it is honest.
Assume the population is initially empty. The terms $U_n(t)$ and $D_n(t)$ are defined as:
$U_n(t)=\sum\limits_{j>n}P_n(t)$ and $D_n(t)=\sum\limits_{j\le n}P_n(t)$
Also define: $U(t)=\lim\limits_{n\to\infty}U_n(t)$ and $D(t)=\lim\limits_{n\to\infty}D_n(t)$.
I have derived a difference equation as $\dfrac{dP_j}{dt}=\lambda_{j-1}P_{j-1}(t)-\lambda_{j}P_j(t)$.
I am asked to show that $\dfrac{dD_n}{dt}=-\lambda_{n}P_n(t)$ and obtain the solution $U_n(t)=\lambda_n\int\limits_{0}^{t}P_n(u)du$.
So \begin{eqnarray} \dfrac{dD_n}{dt}&=&\sum\limits_{j\le n}\frac{dP_j(t)}{dt}\\ &=&\sum\limits_{j\le n}\Big(\lambda_{j-1}P_{j-1}(t)-\lambda_{j}P_j(t)\Big)\\ &=&0-\lambda_1P_{1}(t)+\lambda_1P_1(t)-\lambda_2P_2(t)+\lambda_2P_2(t)\\&-&\lambda_3P_3(t)+\dots+\lambda_{n-2}P_{n-2}(t)-\lambda_{n-1}(t)P_{n-1}(t)+\lambda_{n-1}P_{n-1}(t)-\lambda_nP_n(t).\\ \end{eqnarray}
The first term is zero because $P_{0}(t)=0$, the population is initially empty. Therefore, the terms cancel to leave $\dfrac{dD_n}{dt}=-\lambda_nP_n(t).$
I have no idea how to find the solution.
I have to derive the inequality \begin{equation}U(t)\sum\limits_{j=0}^{n}\dfrac{1}{\lambda_j}\le\int\limits_{0}^{t}D_n(u)du\le\sum\limits_{j=0}^{n}\dfrac{1}{\lambda_j}. \end{equation}
I've no idea about this either.
Finally, show that if $\sum\limits_{j=0}^{\infty}\dfrac{1}{\lambda_j}=\infty$, the process is honest and if $\sum\limits_{j=0}^{\infty}\dfrac{1}{\lambda_j}=C<\infty$, it is dishonest.
I understand that if the birth rate is large enough, then there's a chance of getting an infinite population in a finite time and that's what happens with a dishonest process.
Any help is greatly appreciated, thanks.
