Let $X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n$ be independent binary random variables each being $1$ with probability $\frac{1}{k}$.

Let $Z = X_1(Y_1 + \cdots + Y_k) + X_2(Y_2 + \cdots + Y_{k+1}) + \cdots + X_n(Y_n + Y_1 + \cdots + Y_{k-1})$.

It is easy to see that the expected value of $Z$ is $\mu = \frac{n}{k}$.

Can someone suggest the best possible bound one can obtain for the probability $\Pr[Z \ge (1+\varepsilon) \mu]$?

The setting I am interested in where $n$ is large, and $k \approx \sqrt{n}$.