Description:
<jats:p>In the so-called lightbulb process, on days<jats:italic>r</jats:italic>= 1,…,<jats:italic>n</jats:italic>, out of<jats:italic>n</jats:italic>lightbulbs, all initially off, exactly<jats:italic>r</jats:italic>bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With<jats:italic>X</jats:italic>the number of bulbs on at the terminal time<jats:italic>n</jats:italic>, an even integer, and μ =<jats:italic>n</jats:italic>/2, σ<jats:sup>2</jats:sup>= var(<jats:italic>X</jats:italic>), we have sup<jats:sub><jats:italic>z</jats:italic>∈<jats:bold>R</jats:bold></jats:sub>| P((<jats:italic>X</jats:italic>- μ)/σ ≤<jats:italic>z</jats:italic>) - P(<jats:italic>Z</jats:italic>≤<jats:italic>z</jats:italic>) | ≤<jats:italic>n</jats:italic>Δ̅<jats:sub>0</jats:sub>/2σ<jats:sup>2</jats:sup>+ 1.64<jats:italic>n</jats:italic>/σ<jats:sup>3</jats:sup>+ 2/σ, where<jats:italic>Z</jats:italic>is a standard normal random variable and Δ̅<jats:sub>0</jats:sub>= 1/2√<jats:italic>n</jats:italic>+ 1/2<jats:italic>n</jats:italic>+ e<jats:sup>−<jats:italic>n</jats:italic>/2</jats:sup>/3 for<jats:italic>n</jats:italic>≥ 6, yielding a bound of order<jats:italic>O</jats:italic>(<jats:italic>n</jats:italic><jats:sup>−1/2</jats:sup>) as<jats:italic>n</jats:italic>→ ∞. A similar, though slightly larger bound, holds for odd<jats:italic>n</jats:italic>. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even<jats:italic>n</jats:italic>depends on the construction of a variable<jats:italic>X</jats:italic><jats:sup><jats:italic>s</jats:italic></jats:sup>on the same space as<jats:italic>X</jats:italic>that has the<jats:italic>X</jats:italic>-size bias distribution, that is, which satisfies E[<jats:italic>X</jats:italic><jats:italic>g</jats:italic>(<jats:italic>X</jats:italic>)] = μE[<jats:italic>g</jats:italic>(<jats:italic>X</jats:italic><jats:sup><jats:italic>s</jats:italic></jats:sup>)] for all bounded continuous<jats:italic>g</jats:italic>, and for which there exists a<jats:italic>B</jats:italic>≥ 0, in this case<jats:italic>B</jats:italic>= 2, such that<jats:italic>X</jats:italic>≤<jats:italic>X</jats:italic><jats:sup><jats:italic>s</jats:italic></jats:sup>≤<jats:italic>X</jats:italic>+<jats:italic>B</jats:italic>almost surely. The argument for odd<jats:italic>n</jats:italic>is similar to that for even<jats:italic>n</jats:italic>, but one first couples<jats:italic>X</jats:italic>closely to<jats:italic>V</jats:italic>, a symmetrized version of<jats:italic>X</jats:italic>, for which a size bias coupling of<jats:italic>V</jats:italic>to<jats:italic>V</jats:italic><jats:sup><jats:italic>s</jats:italic></jats:sup>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.</jats:p>