2010/01/05 by Larry Goldstein, Goldstein, Larry, Haimeng Zhang +1
Mathematics · #60B15 #60C05 #62E17 (Primary) #62P10 (Secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1001.0612
openalex publication_date 2010/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(\fracX-μσ ≤ z)-P(Z ≤ z)| ≤ (n)/(2σ2) Δ0 + 1.64 (n)/(σ3)+ \frac2σ where Z is a standard normal random variable, and Δ0 = 1/2√(n) + (1)/(2n) + 1/3 e-n/2 \qmq for n ≥ 6, yielding a bound of order O(n-1/2) as n → ∞. A similar, though slightly larger bound holds for n odd. 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, that satisfies \beas E [X g(X)] =μE[g(Xs)] for all bounded continuous g, \enas and for which there exists a B ≥ 0, in this case B=2, such that X ≤ Xs ≤ X+B almost surely. The argument for n odd is similar to that for n even, 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.