1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

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} $.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.