A Berry-Esseen bound for the lightbulb process

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(where Z is a standard normal random variable and 0 = 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 X s on the same space as X that has the X-size bias distribution, that is, which satisfies E[Xg(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.

Negative dependence and stochastic orderings

“…Goldstein and Zhang [15] show that (at least for n even) W ⋆ ≤ st W + 2, but this is not enough to apply our results directly. Instead, we use the fact, shown by Goldstein and Xia [14], that W is asymptotically distributed as a clubbed binomial distribution.…”

“…This process has also been studied, for example, by Goldstein and Zhang [15], and Goldstein and Xia [14]. See also references therein.…”

“…To prove (2.5), it remains only to show that Λ(W α ) is a decreasing and concave function of α. By equation (15) of [16] we have that…”

Clubbed Binomial Approximation for the Lightbulb Process

“…In particular, the approximation error is less than 1% for n ≥ 21 and less than 0.1% for n ≥ 28. A Berry-Esseen bound in the Kolmogorov metric of order 1/ √ n for the distance between the standardized value of W n and the unit normal was derived in [2]. The lighbulb chain was also studied in [4], and served there as a basis for the exploration of the more general class of Markov chains of multinomial type.…”

Relaxation of monotone coupling conditions: Poisson approximation and beyond

“…and Zhang[13] also provide a coupling such that W ≤ W * ≤ W + 2 a.s., and show that Straightforward calculations (in the spirit of Section 5.3 of[6]) show that D X ≤ α (1−α)k . Hence, letting Y = W + X, the bound of Theorem 6.1 becomes…”