Let $X_i$ be a sequence of independent and identically distributed random variables with $\mathbb{E}(X_i) = \mu$ and var($X_i$) $= 1$ for all $i$. Define $S_n = \frac{1}{n}\sum_{i-1}^n X_i$ and
$$ g(x) = \begin{cases} x & \text{if } x < 1 \\ 2x-1 & \text{if } x \geq 1 \end{cases} $$
Does $\sqrt{n}(g(S_n)-g(\mu))$ converge to anything when $\mu=1$?
I know that for $\mu \neq 1$, I can apply the delta method and obtain convergence in distribution. When $\mu = 1$ it seems like it converges to a "piecewise" distribution. Is there anything to this?
Let $\{X_n\}$ be i.i.d. positive random variables with mean 1 and variance 1. (For example, they can have exponential distribution with parameter 1). By strong law $S_N \to \infty$ almost surely. This implies $g(S_n) \to \infty$ almost surely and $\sqrt {n} (g(S_n)-1) \to \infty$ almost surely.