Suppose we have $\{X_n\}_{n\in\mathbb{N}}$ a sequence of i.i.d. random variables with $\mu=\mathbb{E}[X]<\infty$ and $0<\sigma^2=\text{Var}(X)<\infty$. In this situation, we can apply Levy-Linderberg theorem, which states that $\dfrac{\sum_{i=1}^n X_i-n\mu}{\sqrt{n}\;\sigma}\xrightarrow{d}\ N(0,\;1)$. Applying simple transformation, we can obtain $\frac{1}{n}\sum_{i=1}^n X_i \xrightarrow{d} N\left(\mu,\; \frac{\sigma^2}{n}\right)$
My questions are:
- Can we say that the sequence $\left\{\dfrac{1}{n} \sum\limits_{i=1}^{n} X_i \right\}_{n=1}^{\infty}$ converges to $\mu$ in distribution?
- If $1$ is correct, and $\mu$ is a constant, can we say $\left\{\dfrac{1}{n} \sum\limits_{i=1}^{n} X_i \right\}_{n=1}^{\infty}$ converges to $\mu$ in probability?
- Could we apply Kolmogorov's strong law* and say $\left\{\dfrac{1}{n} \sum\limits_{i=1}^{n} X_i \right\}_{n=1}^{\infty}$ converges to $\mu$ almost surely?
- Can we say anything about converge in r-th mean of $\left\{\dfrac{1}{n} \sum\limits_{i=1}^{n} X_i \right\}_{n=1}^{\infty}$ if we have $X_i \text{ i.i.d } X$, $X\in\text{Exp}(1)$?