发表机构
AlphaNova（AlphaNova）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究横截面信号库的打包与饱和问题，证明仅靠相关性间隔无法保证等权重和的极限，正IC筛选可产生目标对齐但各向同性的二阶矩，且存在指数大库使EWS为PC1或正交于PC1。
AI 中文摘要
我们研究横截面信号库：在每个日期，一个对$d$个资产的预测向量旨在预测下一期的横截面收益。经过去均值和单位归一化后，一个信号是球面上的一个点，其$T$日历史是$T$个球面乘积上的一个点。对历史序列的成对相关性上限是该乘积上的最小角度间隔，因此在这样的上限下增长一个库是一个打包问题。如果一个大型库的成对相关性不太高且等权重，那么随着其增长，等权重和（EWS）是否趋向于一个已知的主成分量？我们在关于库如何填充的明确假设下回答这个问题。仅靠间隔不能保证任何结果：它不固定极限分布；一个饱和的库覆盖球面却可能带有偏倚的计数；在固定域上接近最大打包强制均匀体积，在无限制球面上这给出零均值和无显著主成分（PC1）。对齐取决于准入规则和候选分布。在均匀乘积体积基准下，基于正平均信息系数（IC）的筛选产生非零、目标对齐的均值但各向同性的二阶矩，而正IC边际$\beta$在每个有限$T$下产生一个三水平谱，其主导特征值趋向于$\beta^2$，而残差水平按$1/T$衰减；$T^{-1/2}$阶的边际保持正准入率但特征间隙消失。有限残差谱准则和PC1的分离允许两种结果：存在指数大的正IC库，其EWS是PC1，而其他库的EWS与PC1正交。$\log J=o(T)$足以在$T$个独立日期中对$J$个候选进行均匀估计。推导的结果得到证明并通过数值验证；未使用市场数据。
英文摘要
We study libraries of cross-sectional signals: at each date, a forecast vector over $d$ assets intended to predict the next period's cross-sectional return. Demeaned and unit-normalized, a signal is a point on a sphere and its $T$-date history a point on a product of $T$ spheres. A pairwise correlation cap on histories is a minimum angular separation on that product, so growing a library under such a cap is a packing problem. If a large library is not too pairwise correlated and is equally weighted, does the equally weighted sum (EWS) tend to a known principal-component quantity as it grows? We answer this under explicit assumptions on how the library is filled. Separation alone guarantees nothing: it fixes no limiting distribution; a saturated library covers the sphere yet can carry a biased count; and near-maximum packing on a fixed domain forces uniform volume, which on the unrestricted sphere gives zero mean and no distinguished principal component (PC1). Alignment depends on the admission rule and candidate distribution. Under the uniform product-volume benchmark, screening on positive average information coefficient (IC) yields a nonzero, target-aligned mean but an isotropic second moment, whereas a positive IC margin $β$ makat every finite $T$, with athree-level spectrum whose leading eigenvalue tends to $β^2$ while residual levels decay as $1/T$; margins of order $T^{-1/2}$ keep a positive admission rate but a vanishing eigengA finite residual-spectrum criteS-PC1 alignment.Gilbert-Varshamov codes show separation permits both outcomes: exponentially large positive-IC libraries exist whose EWS is PC1, and others whose EWS is orthogonal to PC1. $\log J=o(T)$ suffices for uniform estimation among $J$ candidates from $T$ iid dates. Derived results are proved and checked numerically; no market data are used.
Comments28 pages, 4 figures, 3 tables. Validation scripts, their outputs, and vector figures are included as ancillary files
