具有对数凹尾的典型过程的主控测度

中文摘要 AI 辅助

设 $Y_1,\ldots,Y_n$ 为具有对数凹尾的独立对称随机变量。我们在坐标尾分布不满足 $\Delta_2$ 条件或正则增长假设的情况下，给出了典型过程 $X_x=\sum_{i=1}^n x_iY_i$ 的期望上确界的维数无关刻画。该刻画由直接由凸尾势决定的尺度依赖内在成本所控制。它既有信息论意义上的率失真表述，也有在原始指标集上的两个等价主控测度表述，其中一个为多尺度形式，另一个基于单一概率测度。更强地，该比较对指标的每个给定分布均成立。特别地，Talagrand 的高斯主控测度定理和 Bednorz-Latała 的伯努利定理可从同一公式分别通过二次成本和截断二次成本恢复。新几何是必要的：一旦移除 $\Delta_2$ 增长条件，基于增量矩的通常链式泛函可能以无界因子超过期望上确界。证明基于对称指数分布的均匀凸序截断原理以及尾斜率的二进分解，该分解同时转移随机过程及其内在成本，且常数是通用的。

英文摘要

Let $Y_1,\ldots,Y_n$ be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process $X_x=\sum_{i=1}^n x_iY_i$ without any $Δ_2$ or regular-growth assumption on the coordinate tails. The characterization is governed by scale-dependent intrinsic costs determined directly by the convex tail potentials. It has an information-theoretic rate-distortion formulation as well as two equivalent majorizing-measure formulations on the original index set, one multiscale and one based on a single probability measure. More strongly, the comparison holds for every prescribed law of the index. In particular, the Gaussian majorizing-measure theorem of Talagrand and the Bernoulli theorem of Bednorz-Latała are recovered from the same formula, through quadratic and truncated quadratic costs, respectively. The new geometry is necessary: once the $Δ_2$ growth condition is removed, the usual chaining functional based on increment moments can exceed the expected supremum by an unbounded factor. The proof is based on a uniform convex-order truncation principle for symmetric exponentials and a dyadic decomposition of the tail slopes that transfers simultaneously the random process and its intrinsic cost, with universal constants.

发表机构

• Shandong University（山东大学）

机构由 AI 辅助整理，请以论文原文为准。