Description:
<p>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, σ 2 = var(X), we have ${\mathrm{sup}}_{\mathrm{z}\in \mathrm{\mathbb{R}}}\left|\mathrm{P}\right((\mathrm{x}-\mathrm{\mu })/\mathrm{\sigma }\le \mathrm{z})-\mathrm{P}(\mathrm{Z}\le \mathrm{z}\left)\right|\le \mathrm{n}{\overline{\mathrm{\Delta }}}_{0}/2{\mathrm{\sigma }}^{2}+1.64\mathrm{n}/{\mathrm{\sigma }}^{3}+2/\mathrm{\sigma }$ , where Z is a standard normal random variable and ${\overline{\mathrm{\Delta }}}_{0}=1/2\sqrt{\mathrm{n}}+1/2\mathrm{n}+{\mathrm{e}}^{-\mathrm{n}/2}/3$ 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[X g (X)] = μ E[ 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.</p>