Advances in Applied Probability

A Berry-Esseen bound for the lightbulb process

Larry Goldstein and Haimeng Zhang

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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 supzR | P((X - μ)/σ ≤ z) - P(Zz) | ≤ nΔ̅0/2σ2 + 1.64n3 + 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 XXsX + 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.

Article information

Source
Adv. in Appl. Probab., Volume 43, Number 3 (2011), 875-898.

Dates
First available in Project Euclid: 23 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.aap/1316792673

Digital Object Identifier
doi:10.1239/aap/1316792673

Mathematical Reviews number (MathSciNet)
MR2858224

Zentralblatt MATH identifier
05955089

Subjects
Primary: 62E17: Approximations to distributions (nonasymptotic) 60C05: Combinatorial probability
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 62P10: Applications to biology and medical sciences

Keywords
Normal approximation toggle switch Stein's method size biasing

Citation

Goldstein, Larry; Zhang, Haimeng. A Berry-Esseen bound for the lightbulb process. Adv. in Appl. Probab. 43 (2011), no. 3, 875--898. doi:10.1239/aap/1316792673. https://projecteuclid.org/euclid.aap/1316792673


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