3
$\begingroup$

If $X_1,X_2,...$ are i.i.d. random variables such that $\phi (t)=E(e^{itX_{1}})=e^{-c|t|^{\alpha }}$, where $c>0$ and $\alpha\in(0,2]$, then show:

  1. For $1<\alpha\leq2$, show that $\frac{X_1+...+X_n}{n}$ converges to $0$ in probability.
  2. For $\alpha=1$, show that $\frac{X_1+...+X_n}{n}$ does not converge to a constant in probability.
  3. For $0<\alpha<1$, show that the distribution of $\frac{X_1+...+X_n}{n}$ does not converge weakly to any probability measure.
  4. For any $\alpha\in(0,2]$, show that there is some $\beta$ such that the distribution of $\frac{X_1+...+X_n}{n^{\beta}}$ does converge to some non-degenerate probability measure.
  • 1
    What did you try? Did you compute the characteristic function of $\frac{X_1+\cdots+X_n}n$? What is the link between the convergence in distribution and the convergence in probability?2011-10-08
  • 0
    The first three parts were listed as properties in a theorem, but a proof was not given. I've been able to prove 1, and have some idea for 2, but am unsure of how to prove 3. Part 4 was given as an exercise after the theorem, and I think $\beta=1/2$ works, but am not positive about that.2011-10-09

1 Answers 1