I’m trying to understand the long‑term growth rate of the total number of balls $G_n$ in the following urn process. I’ve sketched an informal approach using martingale arguments, but I’m not certain how to make each step rigorous.
Problem statement
An urn initially contains one red and one blue ball. In each step $k=1,2,\dots$ we choose a ball uniformly at random from the urn, and then:
- if it is red, we add one red ball and one blue ball (so the total increases by $2$);
- if it is blue, we add one blue ball and $2k+1$ red balls (so the total increases by $2k+2$).
The chosen ball remains in the urn. Let $G_n$ be the total number of balls after $n$ steps. Prove that there exist constants $0<c<\infty$ and $0<\alpha<\infty$ such that
$$
\frac{G_n}{n^\alpha}\longrightarrow c
\quad\text{almost surely.}
$$
My attempt
I write $R_n$ and $B_n$ for the number of red and blue balls after $n$ steps, so $G_n=R_n+B_n$. At time $k$ the probability of drawing a red ball is $R_{k-1}/G_{k-1}$, and of drawing a blue ball is $B_{k-1}/G_{k-1}$. Hence the expected multiplicative increase satisfies
$$
\mathbb{E}\Bigl[\frac{G_k}{G_{k-1}}\Bigm|\mathcal{F}_{k-1}\Bigr]
=\frac{R_{k-1}}{G_{k-1}}\Bigl(1+\frac{2}{G_{k-1}}\Bigr)
+\frac{B_{k-1}}{G_{k-1}}\Bigl(1+\frac{2k+2}{G_{k-1}}\Bigr).
$$
Thus
$$
\mathbb{E}\Bigl[\ln\frac{G_k}{G_{k-1}}\Bigm|\mathcal{F}_{k-1}\Bigr]
\approx
\frac{2\,R_{k-1}+(2k+2)\,B_{k-1}}{G_{k-1}^2},
$$
when $G_{k-1}$ is large. One expects a law of large numbers type result for the sum of these conditional expectations:
$$
\sum_{k=1}^n\mathbb{E}\Bigl[\ln\frac{G_k}{G_{k-1}}\Bigm|\mathcal{F}_{k-1}\Bigr]
\approx\alpha\,\ln n
$$
for some $\alpha>0$. If one can show that the martingale
$$
M_n := \ln G_n - \sum_{k=1}^n\mathbb{E}\Bigl[\ln\frac{G_k}{G_{k-1}}\Bigm|\mathcal{F}_{k-1}\Bigr]
$$
converges almost surely, then
$$
\ln G_n = M_n + \alpha\ln n + o(\ln n)
\;\Longrightarrow\;
\frac{G_n}{n^\alpha}\to e^{M_\infty}=c
\quad\text{a.s.}
$$
So the main steps to fill in are:
- show that
$$ \sum_{k=1}^\infty\mathrm{Var}\bigl(\ln(G_k/G_{k-1})\mid\mathcal F_{k-1}\bigr)<\infty, $$ which guarantees $M_n$ converges a.s. by the martingale convergence theorem; - compute (or at least estimate) the growth rate
$$ \sum_{k=1}^n\mathbb{E}\Bigl[\ln\frac{G_k}{G_{k-1}}\Bigm|\mathcal F_{k-1}\Bigr]\sim\alpha\ln n; $$ - justify the passage from the conditional expectation asymptotics to the almost‑sure limit.
I’d appreciate any guidance on making these martingale arguments rigorous, or alternative proofs (e.g.\ via stochastic approximation) that yield the exact exponent $\alpha$ and the constant $c$.