无全变差收敛的弱收敛的几何性质
On the geometry of weak convergence without total variation convergence
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中文总结 AI 辅助
该研究分析概率测度弱收敛与全变差收敛差异的几何结果，证明过渡集合周长无界增长，给出连通分支数发散的充分条件，通过平面测度序列验证相关结论。
中文摘要 AI 辅助
我们研究概率测度的弱收敛与全变差收敛之间的差异所带来的一些几何结果。考虑定义在$\boldsymbol{R}^d$上、关于勒贝格测度具有密度的概率测度序列，该序列弱收敛于极限测度，但全变差距离始终与极限测度保持有界距离。我们证明，当序列从极限密度下方过渡到上方的集合，其周长（以$(d-1)$维豪斯多夫测度衡量）必须无界增长；且这种增长在固定紧集内依然存在，因此它反映的是这些集合几何复杂性的增加，而非仅在环境空间中的无界扩张。我们进一步给出一个充分条件，在此条件下该集合的连通分支数会发散，这一行为与一维情形极为相似——一维情形中密度序列围绕极限的振荡次数无界增长。我们还通过平面上的一个明确测度序列证明，该条件在一般情形下不可省略：此序列的过渡集合在每一阶段都保持单个连通分支，同时长度和复杂性不断增长；另一个由余弦振荡构造的序列则展示了互补行为，其连通分支数会发散。
英文摘要
We study some geometric consequences of the discrepancy between weak and total variation convergence of probability measures. We consider a sequence of probability measures on $\mathbb R^d$, admitting densities with respect to the Lebesgue measure, that converge weakly to a limiting measure but stay bounded away from it in total variation distance. We show that the sets on which the sequence passes from below to above the limiting density must grow unboundedly in perimeter, as measured by the $(d-1)$-dimensional Hausdorff measure. Moreover, this growth persists within a fixed compact set, so that it must reflect an increase in the geometric complexity of these sets rather than only an unbounded expansion in ambient space. We further provide a sufficient condition under which the number of connected components of the sets diverges, recovering a behavior that is closely reminiscent of the one-dimensional case, in which the number of oscillations of the sequence of densities around the limit grows without bound. We also show that this condition cannot be dispensed with in general, by means of an explicit sequence of measures in the plane whose passing sets remain connected in a single component at every stage while growing in length and complexity. Another sequence, built from cosine oscillations, illustrates the complementary behavior, in which the number of components diverges.
