Actually the question has more details than what it says in the title. Sorry about that I may described the question wrongly.

Let $X_1^n, X_2^n,\dots$ be i.i.d. Bernoulli random variables with parameter $\lambda/n$, i.e. $X_1^n \overset{d}{=}\operatorname{Be}(\lambda/n)$ with fixed $\lambda > 0$. Consider $$ T_i^n := \inf\{k : X_1^n + \cdots + X_k^n = i\}.$$ And I want to show that $$ \frac{T_i^n}{n}\xrightarrow[n\to\infty]{d}\text{Gamma}(i,\lambda).$$

This confuse me since we know that the sum of Bernoulli random variables asymptotically converges to Poisson distribution and I don't see any relationship between Poisson and Gamma distribution.

Can anyone help me out?