A Berry-Esseen bound for the lightbulb process

Abstract

In the so-called lightbulb process, on daysr= 1,…,n, out ofnlightbulbs, all initially off, exactlyrbulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. WithXthe number of bulbs on at the terminal timen, an even integer, and μ =n/2, σ2= var(X), we have supz∈R| P((X- μ)/σ ≤z) - P(Z≤z) | ≤nΔ̅0/2σ2+ 1.64n/σ3+ 2/σ, whereZis a standard normal random variable and Δ̅0= 1/2√n+ 1/2n+ e−n/2/3 forn≥ 6, yielding a bound of orderO(n−1/2) asn→ ∞. A similar, though slightly larger bound, holds for oddn. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for evenndepends on the construction of a variableXson the same space asXthat has theX-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuousg, and for which there exists aB≥ 0, in this caseB= 2, such thatX≤Xs≤X+Balmost surely. The argument for oddnis similar to that for evenn, but one first couplesXclosely toV, a symmetrized version ofX, for which a size bias coupling ofVtoVscan proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.