Joint convergence in distribution and the normalized cycle-character observables

Fix an integer $N\ge 2$. For $n\ge 1$ and $2\le k\le N$ define the real random variable ${\eta }_k^{(n)}(\lambda ):=\frac{p_k^{\mathrm{\#}}(\lambda )}{\sqrt{k}{n}^{k\mathrm{/}2}},\lambda ⊢n,$ on the finite probability space $(Y_n,P_n)$ of The Plancherel measure on the partitions of $n$, whose weights are nonnegative and sum to one by The Plancherel weights sum to one, where $p_k^{\mathrm{\#}}$ is the shifted character observable of Shifted character observables $p_{\rho }^{\mathrm{\#}}$ and profile moments ${\tilde{p}}_k$; since $p_k^{\mathrm{\#}}$ is a real function on $Y_n$, each ${\eta }_k^{(n)}$ is a real random variable on the finite space $Y_n$. Let ${\eta }^{(n)}:=({\eta }_2^{(n)},\dots ,{\eta }_N^{(n)}):Y_n\to {\mathbb{R}}^{N-1}$ be the corresponding ${\mathbb{R}}^{N-1}$-valued random element, with law the pushforward of $P_n$ (Law or distribution of a random element).

Joint convergence in distribution of such vectors means convergence in distribution of random elements in the sense of Convergence in distribution of random elements: if ${\eta }^{(n)}$ has law ${\mu }_n$ and $\eta$ has law $\mu$ on ${\mathbb{R}}^{N-1}$, then ${\eta }^{(n)}⟹\eta$ means ${\mu }_n\Rightarrow \mu$ weakly, that is, $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function $f$ on ${\mathbb{R}}^{N-1}$.

The standard Gaussian target law is $N_{N-1}(0,I_{N-1})$ (Multivariate normal law, including singular covariance), the law of a vector with independent standard normal coordinates; under AC this multi-dimensional law exists and is available in the library. For independent standard Gaussian random variables ${\xi }_k$, $2\le k\le N$, this target is the law of $({\xi }_2,\dots ,{\xi }_N)$. The unnormalized observables $p_k^{\mathrm{\#}}\mathrm{/}{n}^{k\mathrm{/}2}$ instead have the target coordinates ${\zeta }_k:=\sqrt{k}{\xi }_k$, independent centered Gaussians of variances $k$; the normalization ${\eta }_k^{(n)}=p_k^{\mathrm{\#}}\mathrm{/}(\sqrt{k}{n}^{k\mathrm{/}2})$ is chosen so that this is the limit asserted in Kerov's central limit theorem for normalized cycle characters. No convergence is asserted in this definition, and no choice principle is used beyond the existence of the target law.