Let $X_1, X_2,\cdots, X_n $ be a sequence of i.i.d. random variables with common characteristic function $$f(t)=e^{-t^2-\sqrt{|t|}}.$$

$a)$ Find real numbers $\eta_n$ such that $\frac {X_1+ X_2+\cdots+ X_n}{\eta_n}$ converges in distribution to a non- degenerate probability distribution as $n\to \infty$.

$b)$ What possible distributions can the limit have?

$c)$ Prove or disprove $E\left\{\left|X_1\right|\right\}$ is finite.

I've been trying to work on similar lines of the solution here: Symmetric Stable Distribution

But I'm not sure how $e^{-t^2}$ will effect the solution.

I know $\eta_n$ will be of the form $n^{\frac{1}{\alpha}}$ where $\alpha \in (0,2]$.

Also, characteristic function of $\frac {X_1+ X_2+\cdots+ X_n}{\eta_n}$ will be $\lim_n f\left(\frac{t}{n^{\frac{1}{\alpha}}}\right)$.

How to proceed from here?

Edit: My attempt:

$c)$ Since $f(t)$ is not differentiable at $t=0$, therefore $X_1$ is not integrable and hence $E\left\{\left|X_1\right|\right\}$ is not finite.

$a)$ We know, $\eta_n$ is of the form $n^{\frac{1}{\alpha}}$ where $\alpha \in (0,2]$. Let $\alpha = \frac{1}{2}$.

then characteristic function of $\frac {X_1+ X_2+\cdots+ X_n}{\eta_n}$ converges to $e^{-\sqrt{|t|}}$. Hence, $\frac {X_1+ X_2+\cdots+ X_n}{\eta_n}$ converges in distribution to random variable with characteristic function $e^{-\sqrt{|t|}}$.

I'm not sure about the $b)$ part.

  • $\begingroup$ For a): how do you know that the normalization should be of the form $n^{1/a}$? I agree for a). $\endgroup$ – Davide Giraudo Nov 8 '16 at 22:01
  • $\begingroup$ @Davide Giraudo, there is a theorem in Knowing the Odds by John Walsh which says this. $\endgroup$ – Sahiba Arora Nov 8 '16 at 22:12
  • $\begingroup$ OK. Then for part b), you can also consider the normalization $\eta_n=c\cdot n^{1/a}$. $\endgroup$ – Davide Giraudo Nov 9 '16 at 9:23

The distribution of $\sum_{i=1}^nX_i /\eta_n$ is the same as $$\frac{\sqrt n}{\eta_n}N +\frac{n^2}{\eta_n}S, $$
where $N$ and $S$ are two independent random variables of normal and a symmetric stable distribution of parameter $1/2$ respectively. This can be seen from a computation of the characteristic function. Defining $a_n :=\left(\eta_n /\sqrt n\right)^{-1}$, we have to find the possible limits in distribution of the sequence $\left(a_nN +a_n n^{3/2}S \right)_{n\geqslant 1} $.

  • $\begingroup$ Can you check my attempt at the solution? $\endgroup$ – Sahiba Arora Nov 8 '16 at 8:35

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