## Advances in Applied Probability

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

### A Berry-Esseen bound for the lightbulb process

Larry Goldstein and Haimeng Zhang

#### 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, σ^{2} = var(*X*), we have
sup_{z ∈ R} | P((*X* - μ)/σ ≤ *z*)
- P(*Z* ≤ *z*) | ≤ *n*Δ̅_{0}/2σ^{2}
+ 1.64*n*/σ^{3} + 2/σ,
where *Z* is a standard normal random variable and
Δ̅_{0} = 1/2√*n* + 1/2*n* + 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[*X**g*(*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.

#### 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