PomeLemma 18.41.5.44E50E6CAEB1 There is a constant $C_\alpha >0$ such that, for $T\geq 1$,
\[ \mathbb E\bigl |w_{\alpha ,T}(N_ T)-1\bigr |\leq C_\alpha T^{-1/2}. \]
18.41.4 A uniform tail estimateEB75AE06118D
Grouping the monomial degrees by their residues modulo $r$ expresses the kernel through weighted Poisson tails. The estimate of Gamma ratios below removes those weights, with an error independent of the vanishing threshold.
Write $\mathbb P$ and $\mathbb E$ for probability and expectation. For $T>0$, let $N_ T$ be a Poisson random variable with parameter $T$, so that $\mathbb P(N_ T=\ell )=\mathrm{e}^{-T}T^\ell /\ell !$ for integers $\ell \geq 0$. For a real $0<\alpha \leq 1$, define
\[ w_{\alpha ,T}(\ell )=T^{\alpha -1}\frac{\Gamma (\ell +1)}{\Gamma (\ell +\alpha )}. \]
All constants in the next lemma may depend on $\alpha $, but not on $T$.
Proof. Recall that $\mathbb E N_ T=T$ and $\operatorname {Var}(N_ T)=T$. Stirling's formula, applied to the ratio of Gamma functions, gives
\[ \begin{aligned} \frac{\Gamma (\ell +1)}{\Gamma (\ell +\alpha )} & =\ell ^{1-\alpha }(1+O_\alpha (\ell ^{-1}))& & \quad (\ell \to \infty ),\\ \frac{\Gamma (\ell +1)}{\Gamma (\ell +\alpha )} & \leq C_\alpha (\ell +1)^{1-\alpha }& & \quad (\ell \geq 0). \end{aligned} \]
The second bound follows from the first after the constant is increased to account for finitely many small integers. Since $0\leq 1-\alpha <1$, it implies
\[ w_{\alpha ,T}(\ell )\leq C_\alpha \left(1+\frac{\ell +1}{T}\right), \qquad \mathbb E(w_{\alpha ,T}(N_ T)+1)^2\leq C_\alpha \quad (T\geq 1). \]
For $T\geq 2$, consider the event $E_ T=\{ |N_ T-T|\leq T/2\} $. On this event, the expansion of the Gamma ratio and the bounded derivative of $s\mapsto s^{1-\alpha }$ on $[1/2,3/2]$ give
\[ |w_{\alpha ,T}(N_ T)-1| \leq C_\alpha \left(\frac{|N_ T-T|}{T}+\frac1T\right). \]
By the Cauchy–Schwarz inequality, its expectation is at most $C_\alpha T^{-1/2}$. On the complement, Chebyshev's inequality gives $\mathbb P(E_ T^ c)\leq 4/T$, and the preceding second-moment bound together with the Cauchy–Schwarz inequality gives the same estimate for the contribution of $E_ T^ c$. Enlarging the constant handles $1\leq T\leq 2$. $\square$
For integers $q\geq 0$ and $0\leq s<r$, put
\[ L_ s(q)=\left\lceil \frac{q-s}{r}\right\rceil \geq 0. \]
If a monomial degree is written as $k=r\ell +s$, the condition $k\geq q$ becomes $\ell \geq L_ s(q)$.
Proposition 18.41.6.79BA8CB51C0E There is a constant $C_ r>0$, depending only on $r$, such that for $m>0$, $z\ne 0$ and $T=m|z|^{2r}\geq 1$,
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18.41.7
\begin{equation} \left|\frac{\pi B_{m,r}^{(q)}(z)}{mr^2|z|^{2r-2}} -\frac1r\sum _{s=0}^{r-1}\mathbb P\{ N_ T\geq L_ s(q)\} \right| \leq C_ r T^{-1/2} \quad (q\geq 0). \end{equation}
In particular, the error is uniform in the vanishing order $q$.
Proof. Let $\rho =|z|>0$ and put $\alpha _ s=(s+1)/r$ for integers $0\leq s<r$. Grouping the positive series (18.41.2) by the residues $s$ and using $T=m\rho ^{2r}$, we obtain the exact identity
\[ \begin{split} B_{m,r}^{(q)}(z) & =\frac r{\pi \rho ^2}\sum _{s=0}^{r-1} \mathrm{e}^{-T}\sum _{\ell \geq L_ s(q)} \frac{T^{\ell +\alpha _ s}}{\Gamma (\ell +\alpha _ s)}\\ & =\frac{rT}{\pi \rho ^2}\sum _{s=0}^{r-1} \mathbb E\bigl [w_{\alpha _ s,T}(N_ T) \mathbf1_{\{ N_ T\geq L_ s(q)\} }\bigr ]. \end{split} \]
Divide by $mr^2\rho ^{2r-2}/\pi =r^2T/(\pi \rho ^2)$ and apply lemma 18.41.5 to each of the finitely many residues. Since the indicators are bounded by one independently of their thresholds, the resulting error is uniform in $q$. $\square$
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