论文标题
二次混乱的波动
Fluctuations of Quadratic Chaos
论文作者
论文摘要
在本文中，我们表征随机二次形式的所有分布限制$ t_n = \ sum_ {1 \ le u <v \ le n} a_ {u，v} x_u x_v $，其中$（a_ {a_ {u，v}）对角线上的零和$ x_1，x_2，\ ldots，x_n $是I.I.D.特别是，我们表明，$ s_n的任何分布限制：= t_n/\ sqrt {\ mathrm {var} [t_n]} $都可以表示为三个独立组件的总和：高斯，一个（可能是）独立的中心chi-squares和与随机的无数级别混合物的无限加权sum和随机的随机变量。结果，我们证明了$ s_n $的渐近正态性的第四刻定理，即使$ f $没有有限的第四刻，它也适用。更正式地，我们显示$ s_n $在$ n（0，1）$时收敛，并且仅当$ s_n $的第四刻（当$ f $没有有限的第四刻时适当截断）收敛到3（标准正态分布的第四刻）。
In this paper we characterize all distributional limits of the random quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u, v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal and $X_1, X_2, \ldots, X_n$ are i.i.d.~ mean $0$ variance $1$ random variables with common distribution function $F$. In particular, we show that any distributional limit of $S_n:=T_n/\sqrt{\mathrm{Var}[T_n]}$ can be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite weighted sum of independent centered chi-squares, and a Gaussian mixture with a random variance. As a consequence, we prove a fourth moment theorem for the asymptotic normality of $S_n$, which applies even when $F$ does not have finite fourth moment. More formally, we show that $S_n$ converges to $N(0, 1)$ if and only if the fourth moment of $S_n$ (appropriately truncated when $F$ does not have finite fourth moment) converges to 3 (the fourth moment of the standard normal distribution).
