A Berry-Esseen bound for the lightbulb process

In the so-called lightbulb process, on daysr= 1,…,n, out ofnlightbulbs, all initially off, exactlyrbulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. WithXthe number of bulbs on at the terminal timen, an even integer, and μ =n/2, σ2= var(X), we have supz∈R| P((X- μ)/σ ≤z) - P(Z≤z) | ≤nΔ̅0/2σ2+ 1.64n/σ3+ 2/σ, whereZis a standard normal random variable and Δ̅0= 1/2√n+ 1/2n+ e−n/2/3 forn≥ 6, yielding a bound of orderO(n−1/2) asn→ ∞. A similar, though slightly larger bound, holds for oddn. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for evenndepends on the construction of a variableXson the same space asXthat has theX-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuousg, and for which there exists aB≥ 0, in this caseB= 2, such thatX≤Xs≤X+Balmost surely. The argument for oddnis similar to that for evenn, but one first couplesXclosely toV, a symmetrized version ofX, for which a size bias coupling ofVtoVscan 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.

A Berry-Esseen bound for the lightbulb process

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 supz ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 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 X ≤ Xs ≤ 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 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.

Precise Rates in Log Laws for NA Sequences

LetX1,X2,…be a strictly stationary sequence of negatively associated (NA) random variables withEX1=0, setSn=X1+⋯+Xn, suppose thatσ2=EX12+2∑n=2∞EX1Xn>0andEX12<∞,if−1<α≤1;EX12(log|X1|)α<∞, ifα>1. We provelimε↓0ε2α+2∑n=1∞((logn)α/n)P(|Sn|≥σ(ε+κn)2nlogn)=2−(α+1)(α+1)−1E|N|2α+2, whereκn=O(1/logn)and N is the standard normal random variable.

On an inequality for the normal distribution arising in bioequivalence studies

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

On an inequality for the normal distribution arising in bioequivalence studies

For (μ,σ2) ≠ (0,1), and 0 &lt; z &lt; ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 &lt; z &lt; ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

Asymptotic Approximation of an Integral Involving the Normal Distribution*

An asymptotic approximation is obtained, as k → ∞, for the integralwhere Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.