I am reading "Introduction to Stochastic Processes" by Lawler, and I am having trouble understanding his intuitive explanation of a CLT for renewal processes (Chapter 6, p.134).
Here is his argument.
Let $T_1,T_2,\dots$ be iid nonnegative random variable. Let $\mu=E[T_i]$ and $\sigma^2 = V(T_i)$. Define the renewal process by $N_t=\max\{n\mid T_1+\dots T_n\le t\}$. The usual CLT implies that, roughly speaking, $T_1+\cdots T_n$ follows the same distribution as $n \mu + \sigma \sqrt{n}B$, where $B$ is follows the standard normal. This states that the number of occurrences in time $n\mu+\sigma\sqrt{n} B$ is $n$. From the law of large numbers, $N_t/t\to\mu^{-1}$, so the number of occurrences in the time interval of size $\sigma \sqrt{n}|B|$ is about $\sigma \sqrt{n} |B|/\mu$. Hence, we have the number of occurrences in time $n\mu$ is about
$$ n -\frac{\sigma}{\mu}\sqrt{n}B. $$
I understand the arguments except the last sentence. I can make sense of this if the last expression is $n-\sigma/\mu\sqrt{n}B$ instead. Since the number of occurrences in a unit time is $1/\mu$ by the LLN, in time $n\mu+\sigma\sqrt{n}B$, we expect $$ (n\mu+\sigma\sqrt{n}B) \times 1/\mu=n +\frac{\sigma}{\mu}\sqrt{n}B. $$
What am I missing? He proceeds to use the fact that $B$ is symmetric and replace $t$ with $n\mu$ to get $N_t \approx\mu^{-1}t+\mu^{-3/2}\sigma\sqrt{t}B$ and derive the CLT. I'm not looking for a formal proof of the CLT but rather trying to make sense of his explanation.