Show that $(X_1 + \cdots + X_n )/n \to 0$ in distribution by characteristic function

characteristic-functionsprobability theoryweak-convergence

There's a random variable $X$ whose characteristic function is $\phi(t) = E(e^ {itX})$ is given by $$\phi(t) = e^{-|t|^{3/2}}$$ Let $X_1, X_2, \ldots$ the independent random variable with the same distribution as $X$.

  1. Show that $(X_1 + \cdots + X_n )/n \to 0$ in distribution.

  2. Find a choice of constants $C_n$, depending on $n$ , such that $(X_1 + \cdots + X_n )/C_n \to X$ in distribution.

What I tried with continuity theorem:
but I don't know how to continue the next steps. Thank you!

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Best Answer

Pay attention to exponent rules. The characteristic for the sum divided by $n$ is. $$({e^{-(|\frac{t}{n}|^{3/2})}})^n={e^{-n(|\frac{t}{n}|^{3/2})}}={e^{-\frac{1}{\sqrt n}(|t|^{3/2})}}$$

This converges to $1$, which is the characteristic of $0$.

Step 2, we need

$${e^{(C(n))^{-3/2}n(|t|^{3/2})}}=e^{-(|t|^{3/2})}$$

Hence $C(n)=n^{2/3}$ from some $n>k$.

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