A Berry Esseen Theorem for the Lightbulb Process

In the so called lightbulb process, on days $r=1,...,n$, out of $n$ lightbulbs, all initially off, exactly $r$ bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With $X$ the number of bulbs on at the terminal time $n$, an even integer, and $\mu=n/2, \sigma^2={Var}(X)$, we have $\sup_{z \in \mathbb{R}} |P(\frac{X-\mu}{\sigma})-P(Z \le z)| \le \frac{n}{2\sigma^2}\Delta_0 + 1.64 \frac{n}{\sigma^3}+ \frac{2}{\sigma}$ where $$ \Delta_0 \le \frac{1}{2\sqrt{n}} + \frac{1}{2n} + e^{-n/2} \qmq {for $n \ge 4$,} $$ yielding a bound of order $O(n^{-1/2})$ as $n \to \infty$. A similar, though slightly larger bound holds for $n$ odd. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even $n$ depends on the construction of a variable $X^s$ on the same space as $X$ which has the $X$ size bias distribution, that is, that satisfies E X g(X)=\mu Eg(X^s) \quad for all bounded continuous $g$, and for which there exists a $B \ge 0$, in this case B=2, such that $X \le X^s \le X+B$ almost surely. The argument for $n$ odd is similar to that for $n$ even, but one first couples $X$ closely to $V$, a symmetrized version of $X$, for which a size bias coupling of $V$ to $V^s$ can proceed as in the even case.