PomeLemma 18.41.5.44E50E6CAEB1 There is a constant $C_\alpha >0$ such that, for $T\geq 1$,
\[ \mathbb E\bigl |w_{\alpha ,T}(N_ T)-1\bigr |\leq C_\alpha T^{-1/2}. \]
Lemma 18.41.544E50E6CAEB1
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 $1\leq T\leq 2$. $\square$
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