A Berry-Esseen bound for the lightbulb process

Abstract

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 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, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n−1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. 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 Xs on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ Xs ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to Vs can 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.