Proposition 18.44.8.443408C4E29A Keep $\alpha _ j,\tau ,R,T_ m$ as above, but use instead the homogeneous radii given by $t_ j=(\alpha _ jT_ m/(r_ jm))^{1/r_ j}$, and put $r_*:=\max _ jr_ j$. For the metrics of this section, the law
\[ \frac{\pi ^ qB_ m(z)}{m^ q\prod _ jr_ j^2|z_ j|^{2r_ j-2}}\longrightarrow \Phi (\tau ) \]
holds for every fixed $\tau $ if and only if $k_ m=o(m^{2/(r_*+2)})$. If $k_ m/m^{2/(r_*+2)}\to c\in (0,\infty )$, the left side instead tends to
\[ \Phi \bigl (\tau +C_\alpha c^{(r_*+2)/(2r_*)}\bigr ),\qquad C_\alpha =\frac1{\sqrt R}\sum _{j:r_ j=r_*} (r_*+1)\varepsilon _ j(\alpha _ j/r_*)^{1+1/r_*}. \]
If that ratio tends to infinity while $k_ m=o(m)$, the limit is $1$.
Proof.
By (18.44.1), $\sum _ j\mu _ j=T_ m+\Delta _ m$ with
\[ \Delta _ m=\sum _ j(r_ j+1)\varepsilon _ jmt_ j^{r_ j+1} +O\left(\sum _ jmt_ j^{2r_ j}\right),\qquad V\sim Rk_ m. \]
Since $r_ j\geq 2$, each error term is negligible compared with the corresponding positive correction. Consequently
\[ \frac{\Delta _ m}{\sqrt{Rk_ m}}=(1+o(1))\sum _ j \frac{(r_ j+1)\varepsilon _ j(\alpha _ j/r_ j)^{1+1/r_ j}}{\sqrt R} k_ m^{1/2+1/r_ j}m^{-1/r_ j}. \]
At the critical scale only the indices with $r_ j=r_*$ contribute in the limit. Below it every term tends to zero, and above it a positive term diverges. The Gaussian limit of the normalized sum proved above applies in all three cases; in the last case its tightness gives probability one. Moreover, $a(t)\to 1$ and $g_ j(t_ j)/(r_ j^2t_ j^{r_ j-1})\to 1$, so the two normalizations agree. If $k_ m/m^{2/(r_*+2)}$ does not tend to zero, a subsequence has a positive finite or infinite limit, and the law fails at $\tau =0$. This proves both directions of the equivalence for these metrics.
$\square$
Pome
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