In the so called light bulb process of Rao, Rao and Zhang (2007), on days r =
1, . . . , n, out of n light bulbs, 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 = n/2, σ2 =
varX, we have
sup
∈R 􏰐
􏰐
P ( X −
σ ≤ z ) − P (Z ≤ z )
􏰐􏰐 ≤
n
2σ2 ∆0 + 1.64
n
σ3 +
2
σ
where Z is a
N (0, 1) random variable and
∆0
≤
1
2√n +
1
2n + e−
n/2
, for n
≥ 4,
yielding a bound of order O(n−1/2 ) as n
→ ∞.
The results are shown using a version of Steins 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 which has the X size bias distribution, that is, which
satisfies
E[X g(X )] = 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 n even, 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.
