arXiv 1001.06122010-01math.PRmath.STstat.TH

灯泡过程的Berry Esseen定理

A Berry Esseen Theorem for the Lightbulb Process

Larry Goldstein, Haimeng Zhang

备注：: 38 pages - Version 3 provides a much shorter and completely transparent proof of Lemma 3.2. An error in the odd case coupling has been corrected, resulting in a somewhat larger constant, and a simpler overall expression for the bound. The constant for the even case has been slightly improved

摘要：

在所谓的灯泡过程中，在$r=1,..., n$日，在$n$灯泡之外，所有最初关闭的、确切地说是$r$灯泡，被统一地选择并且独立于过去，使它们的状态从关闭改变为打开，或者反之亦然。对于$X$，即在终端时间打开的灯泡数$n$，一个偶数，以及$μ=n/2, σ^2=Var(X)$，我们有$$ \sup_{z \in \mathbb{R}} |P(\frac{X-μ}σ \le z)-P(Z \le z)| \le \frac{n}{2σ^2} \barΔ_0 + 1.64 \frac{n}{σ^3}+ \frac{2}σ $$，其中$Z$是一个标准正态随机变量，而$$ \barΔ_0 = 1/2\sqrt{n}} + \frac{1}{2n} + 1/3 e^{-n/2} \qmq {for $n \ge 6$,} $$产生一个阶为$n \to \infty$的$O(n^{-1/2})$的界。一个类似的，但稍大的界限适用于$n$奇数。结果显示，使用一个版本的斯坦的方法有界的，单调的大小偏置耦合。偶数$n$的论证取决于在与$X$相同的空间上构造一个变量$X^s$，该变量具有$X$ -大小偏差分布，即对于所有有界连续$g$，\enas，满足\beas E [X g（X）] =μE[g（X^s）] \quad，并且存在一个$B \ge 0$，在这种情况下B=2，使得$X \le X^s \le X+B$几乎必然。$n$奇数的论证类似于$n$偶数的论证，但是首先将$X$紧密地耦合到$V$，这是$X$的对称化版本，对于$V$到$V^s$的尺寸偏差耦合可以像在偶数情况下一样进行。在偶数和奇数情况下，条件期望方差的关键计算都需要灯泡链谱分解的详细信息。

英文摘要：

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 $$ \sup_{z \in \mathbb{R}} |P(\frac{X-μ}σ \le z)-P(Z \le z)| \le \frac{n}{2σ^2} \barΔ_0 + 1.64 \frac{n}{σ^3}+ \frac{2}σ $$ where $Z$ is a standard normal random variable, and $$ \barΔ_0 = 1/2\sqrt{n}} + \frac{1}{2n} + 1/3 e^{-n/2} \qmq {for $n \ge 6$,} $$ yielding a bound of order $O(n^{-1/2})$ as $n \to \infty$. A similar, though slightly larger bound holds for $n$ odd. 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, that satisfies \beas E [X g(X)] =μE[g(X^s)] \quad for all bounded continuous $g$, \enas and for which there exists a $B \ge 0$, in this case B=2, such that $X \le X^s \le X+B$ almost surely. The argument for $n$ odd 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. 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.

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摘要：

英文摘要：

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