= Squared-population near-invariant for stochastic attrition
{title2=$D_j=A_j^2-B_j^2,\quad\mathbb E(\Delta D_j\mid\mathcal F_{j-1})=\frac{B_{j-1}-A_{j-1}}{A_{j-1}+B_{j-1}}$}
In the <jump chain> of the <symmetric stochastic attrition process>, the two possible increments of $D$ are $-2A+1$ and $2B-1$. Their mean is the displayed quantity, bounded by one in absolute value. Subtracting these conditional means produces a <martingale>. If both initial populations are at most $N$, it has at most $2N$ nonzero increments, each bounded by $4N$. Thus <Azuma-Hoeffding inequality> controls changes of order $N^2$ with exponentially small probability of order $e^{-cN}$.
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