= Exponential concentration of terminal stochastic attrition survivors
{title2=$\mathbb P(|X^N-\sqrt{1-\alpha^2}|>\delta)\leq2e^{-(1-\alpha^2)\delta^2N/256}$}
Start the <symmetric stochastic attrition process> at $(N,\lfloor\alpha N\rfloor)$, with $0<\alpha<1$, and let $X^N$ be the final first-population size divided by $N$. The bound holds for all sufficiently large $N$, depending on $\alpha,\delta>0$. At absorption, one coordinate is zero. If the first survives, the terminal squared difference is $N^2(X^N)^2$; if it is extinguished, that difference is nonpositive. In either case the indicated deviation forces an error of at least $\sqrt{1-\alpha^2}\,\delta N^2$ from the limiting squared invariant. The initial rounding error and total conditional drift are only $O(N)$. Apply <Azuma-Hoeffding inequality> to the centered <squared-population near-invariant for stochastic attrition>, using its $2N$ increments bounded by $4N$, to obtain the displayed exponent.
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