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 sup_{z ∈ 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 X^s on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(X^s)] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ X^s ≤ 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 V^s 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.