Description:
There is an indicator function called $I_{i}^{k}$ as follows:
\begin{align*}
\begin{split}
I_{i}^{k}= \left\{
\begin{array}{lr}
1, \; {\rm if \; the \; event \; happened\; at\; time \;k\; for\; player\; i;}\\
0, \; {\rm if \; not.}
\end{array}
\right.
\end{split}
\end{align*}
The event happened at least once, so $\mathbb{E}[I_{i}^{k}]=p_{i}>0$. Besides, the indicators
$I_{i}^{k}, \left\lbrace i=1,\dots N \right\rbrace $ are independent and
identically distributed (i.i.d.) with a fixed distribution across
time steps.
Now, we Let $\Gamma_{i}^{k}=\sum_{t=1}^{k}I_{i}^{t}$ and $\gamma_{i,k}={1}/{(\Gamma_{i}^{k})^{a}}$, $a\in (1/2, 1]$ for all $i$ and $k\geq 1$.
Can we give the upper bound of the following?
\begin{align*} (a) \sum_{k=1}^{K}\mathbb{E}\left[ \gamma_{i,k}^{2}\right] \leq ? \quad\quad (b) \sum_{k=1}^{K}\mathbb{E}\left[ \left| \gamma_{i,k}-\frac{1}{k^{a}p_{i}^{a}} \right| \right] \leq ? \end{align*} where $K$ is some constant.
PS: I think maybe it can be proved by Taylor expansion, but I don't know how to calculate it, maybe the following link is useful: Expectation of inverse of sum of positive iid variables
