# Stein's Method for the Lightbulb Process (Larry Goldstein and Haimeng Zhang)

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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 o&#64256;, exactly r bulbs, selected uniformly and

independent of the past, have their status changed from o&#64256; to on or vice versa. With

X the number of bulbs on at the terminal time n, an even integer and = n/2, &#963;2 =

varX, we have

sup

&#8712;R &#1113104;

&#1113104;

P ( X &#8722;

&#963; &#8804; z ) &#8722; P (Z &#8804; z )

&#1113104;&#1113104; &#8804;

n

2&#963;2 &#8710;0 + 1.64

n

&#963;3 +

2

&#963;

where Z is a

N (0, 1) random variable and

&#8710;0

&#8804;

1

2&#8730;n +

1

2n + e&#8722;

n/2

, for n

&#8805; 4,

yielding a bound of order O(n&#8722;1/2 ) as n

&#8594; &#8734;.

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

satis&#64257;es

E[X g(X )] = E[g(X s )], for all bounded continuous g

and for which there exists a B

&#8805; 0, in this case, B = 2, such that X &#8804; X

s

&#8804; X + B

almost surely. The argument for odd n is similar to that for n even, but one &#64257;rst

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.