Lower bound on the probability of a sum of iid random variables larger than a threshold

Question

Suppose you are given $n$ iid random variables $X_1,\ldots,X_n$ each taking values in $\{1,2,3,\ldots\}$. Each $X_i$ has the cumulative distribution function $P(X_i \le v) = 1 - \frac{1}{(v+1)^p}$ for some $p \ge 3$. I am interested in lower bounding the probability $P(X_1 + X_2 + \ldots + X_n \ge t)$. In particular, is it possible to get a constant lower bound on the probability for $t = n^{\gamma}$ for some $\gamma > 1$. In case this is not possible, I would also appreciate a counter-example e.g. for $p=3$.