According to the Weak Law of Large Numbers (WLLN), we have that X (n) /n converges to p in probability. Hence, (1 + X (n) )/n converges to p in probability and since f (x) = x α is continuous on (0, ∞), it is verified that n -α (1 + X (n) ) α converges to p α in probability. At this point, similarly, we are going to verify that {((1 + X (n) 1 )/n) α } n is uniformly integrable. It is worth noting that for α ≥ 0, verifying the statement is straightforward as (1 + X (n) )/n ≤ 1, for all n. Let's check the case α < 0: To see this, let ε ∈ (0, p) and define