A Stability of Symmetrization of Product Measures with Few Distinct Factors

Abstract

Let $\mathscr{P} = \{F_0,\cdots, F_m\}$ be a class of probability measures on $(\mathscr{X}, \mathscr{B})$. For any signed measure $\tau$ on $\mathscr{B}^N$, let $\tau^\ast$ be the average of $\tau g$ over all $N$! permutations $g$ and let $\|\tau\| = \mathbf{V}\{|\tau(C)|: C \in \mathscr{B}^N\}$. Let $d_{ij} = \|F_i - F_j\|$ and $K(x) = .5012\cdots x(1 - x)^{-\frac{3}{2}}$. For any nonnegative integral partitions $\mathbf{N} = (N_0,\cdots, N_m)$ and $\mathbf{N}' = (N_0',\cdots, N_m')$ of $N$, let $\delta_i = N'_i - N_i$ and $\wedge_i = (N'_i \wedge N_i) + 1$. With $\tau = \times F_i^{N_i} - \times F_i^{N_i'}$ and $n = {\tt\#}\{i\mid\delta_i \neq 0\} - 1$, we bound $\|\tau^\ast\|^2$ by \begin{equation*}\tag{T3} nK(d)\sum \delta_i^2\wedge_i^{-1}\quad \text{with} d = \vee\{d_{ij}\mid\delta_i \neq 0, \delta_j \neq 0\}\end{equation*} and, if $\mathscr{P}$ is internally connected by chains with non-orthogonal successive elements, by \begin{equation*}\tag{T4} \frac{1}{2}mK(\check{d})(\sum |\delta_i|)^2(\sum \wedge_i^{-1}) \quad\text{with} \check{d} = \vee\{d_{ij} \mid F_i ?? F_j\}.\end{equation*} The bound (T3) is finite iff the $F_i$ with $\delta_i \neq 0$ are pairwise non-orthogonal and (T4) is designed to replace it otherwise.