Proving a sum to be sublinear in growth

Question

Suppose that $\alpha \in (0,1)$. The goal is to prove that the following sum is of $o(T)$ (or, if possible, give a more accurate growth rate, e.g. $O(T^{1-\alpha})$ or something like that): $$ \sum_{t=1}^T \prod_{i=0}^{\lfloor\sqrt{t}\rfloor-1}(1 - (t-i)^{-\alpha}).$$ It is also easy to see that the growth rate of the sum below is higher than the one above. So proving sub-linearity of the sum below also is enough: $$ \sum_{t=1}^T (1 - t^{-\alpha})^{\sqrt{t}}.$$

Answer 1

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\R}{\mathbb{R}} \newcommand{\F}{\mathcal{F}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\PP}{\operatorname{\mathsf P}} \newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$

For $i=0,\dots,\lfloor\sqrt{t}\rfloor-1$, we have $t-i=ct$; here and in what follows $c$ denotes various positive quantities (which possibly differ from one another even in the same formula) bounded away from $0$ and $\infty$ over all \begin{equation} \al\in(0,1),\quad T=1,2,\dots,\quad t=1,\dots,T,\quad i=0,\dots,\lfloor\sqrt{t}\rfloor-1;\tag{1} \end{equation} that is, there is a universal positive real constant $C$ such that for each of these instances of $c$ and for all $\al$, $T$, $t$, and $i$ as in (1) one has $1/C\le c\le C$. So, the sum in question is \begin{equation*} s_T:=\sum_{t=1}^T (1 - (ct)^{-\alpha})^{c\sqrt{t}} =\sum_{t=1}^T \exp\{-ct^{1/2-\al}\}. \end{equation*}

If $\al\ge1/2$, then $\exp\{-ct^{1/2-\al}\}=c$ and hence $s_T$ is $cT$ and not $o(T)$.

If $\al<1/2$, then $\exp\{-ct^{1/2-\al}\}=o(1)$ as $t\to\infty$ and hence $s_T=o(T)$ as $T\to\infty$; in this case, we even have $s_T=O(1)$.