Theorem: Uniform-tail joint-limit criterion

Let $v_{\delta ,R}\in {\mathcal{D}}^{\prime }(X)$. Suppose that for each fixed sufficiently large $R$,

and, for every compact test support $H$, there are $C_H,j,\beta >0$ such that

uniformly for sufficiently small $\delta$. Then $T_R$ converges strongly to a distribution $T$, and

jointly as $\delta \downarrow 0$ and $R\to \infty$.

Proof. Pass to the fixed-$R,R^{\prime }$ limits in (eq:uniform-tail). The same bound holds for $T_{R^{\prime }}-T_R$. It makes the family Cauchy on each compact test-support class.

An anchor $T_{R_0}$, together with the tail bound, gives a finite-order compact-local bound for the limit, so the limit is a distribution. The estimate is uniform on bounded test sets and gives strong convergence.

For joint convergence, fix a bounded test set and choose $R_0$ so its tail bound is small. For all $R\ge R_0$, compare $v_{\delta ,R}$ to $v_{\delta ,R_0}$. The latter converges at fixed $R_0$, and $T_{R_0}$ is close to $T$. The three differences prove the joint limit.