Strong Law of Large Number for a process

Question

Given a sequence $\{X_n\}$ of iid non-negative (like exponential) r.v.'s with expectation $1$, we know from SLLN $$\sum_{n=1}^k X_n / k \rightarrow 1 \quad \text{ a.e. }.$$ Suppose $X_n$ is the wait time for the $n$th package to arrive, we can define a integer valued process $f(t)$ which tells us how many packages have arrived up to time $t$ $$f(t): =\max\left\{k : \sum_{n=1}^k X_n < t \right\}.$$ How can I show that $$\lim_{t\rightarrow \infty} f(t)/t = 1? $$ We know $\{f(t) < k\} = \left\{\sum_{n=1}^k X_n > t\right\}$. And intuitively I know $f(t)/t$ is just like $\frac{k}{\sum_{n=1}^k X_n}$ which goes to $1$. How can I make this rigorous.

Answer 1

(Note: There's a typo in your definition of $f$; you should write $\max$ instead of $\min$.)

Your $f(t)$ is an example of a renewal process.

To simplify notation, I write $T_k=\sum_{n=1}^kX_n$. By definition, we have $T_{f(t)}<t\leq T_{f(t)+1}$. Then $$\frac{T_{f(t)}}{f(t)}<\frac{t}{f(t)}\leq\frac{T_{f(t)+1}}{f(t)+1}\frac{f(t)+1}{f(t)},$$ so (since $f(t)\to\infty$ pointwise and we have by the SLLN that the left and right-hand sides converge to $\mu$ almost everywhere) $\lim_{t\to\infty}t/f(t)=1$ a.e. Then, by nice properties of limits, we have $$\lim_{t\to\infty}f(t)/t=1$$ a.e.

In general, the result $\lim_{t\to\infty}f(t)/t=1/\mu$ holds for any $\mu>0$. (By $\mu$, I mean the expectation of the $X_n$'s.) For reference, the "Asymptotic properties" section of the renewal theory Wikipedia page and pp.126-127 of Durrett's Essentials of Stochastic Processes (2016) have very similar proofs.