I was posed the following question and I am interested in whether or not my proposed approach works.
Let $X_{i}$ be i.i.d uniform random variables on $[0,1]$ and define $N_{k}$ to be the smallest integer such that $$1<\sum_{i=1}^{N_{k}} X^{k}_{i}.$$
Then calculate $$\lim_{k\rightarrow\infty}\frac{\mathbb{E}[N_{k}]}{k}.$$
My approach is to call $N_{k}$ a stopping time for the sequence $X_{1},X_{2},\dots$ so that
$$X_{1}^{k}+X_{2}^{k} + \dots + X_{N_{k}-1}^{k} \leq 1 < X_{1}^{k}+\dots + X_{N_{k}}^{k}.$$
Then using Wald's equation and taking the expectation of each side,
$$\mathbb{E}[N_{k}-1]\mathbb{E}[X_{1}^{k}] \leq 1 < \mathbb{E}[N_{k}]\mathbb{E}[X_{1}^{k}] $$ or $$\mathbb{E}[N_{k}]\mathbb{E}[X_{1}^{k}] -\mathbb{E}[X_{1}^{k}] \leq 1 \leq \mathbb{E}[N_{k}]\mathbb{E}[X_{1}^{k}]$$
$$-\mathbb{E}[X_{1}^{k}] \leq 1 - \mathbb{E}[N_{k}]\mathbb{E}[X_{1}^{k}] \leq 0$$
and conclude via the squeeze theorem by noting that
$$\lim_{k\rightarrow \infty} \frac{\mathbb{E}[X_{1}^{k}]}{k} = 0, \quad\lim_{k\rightarrow \infty} \mathbb{E}[X_{1}^{k}] = 1.$$
Is this general approach going to be successful? I am not too confident in taking the expectation of the inequality because the line
$$X_{1}^{k}+X_{2}^{k} + \dots + X_{N_{k}-1}^{k} \leq 1 < X_{1}^{k}+\dots + X_{N_{k}}^{k}$$
seems to be abusing notation. I think it is conditioning on $N_{k}$ satisfying this inequality and something subtle might be making this argument invalid.