Limit probability of a tricky random variable

Question

Assume that $N_k, k = 1,2,\ldots$ are independent random variables distributed as $\mathrm{Bin}(k,p)$, respectively. Let $X_1,X_2,\ldots$ be independent $\mathrm{Exp}(a)$-distributed random variables of $N_k, k = 1,2,\ldots$

Find the limit in probability of $Y_k = \dfrac{ \sum_{i=1}^k (X_i-a)}{N_k}$ as $k \rightarrow \infty $

All the examples I've found on this kind of questions have been pretty simple. But this one I can't figure out how to deal with.

Answer 1

Hint :

$Y_k = \dfrac{ \sum_{i=1}^k (X_i-a)}{N_k}=\dfrac{ \sum_{i=1}^k (X_i-a)} k \dfrac k {N_k}$

And $N_k=\sum_{i=1}^k Z_i$ with $Z_i \sim B(1,p)$ (A Bernouilli variable).

Then use the weak version of a famous theorem in probability...