Prove that, no matter what the constants (alpha>0) and (beta>0), [ varphi(t)=left(1+frac{t^{2}}{beta^{2}} ight)^{-alpha} ] is an infinitely

Prove that, no matter what the constants \(\alpha>0\) and \(\beta>0\),

\[ \varphi(t)=\left(1+\frac{t^{2}}{\beta^{2}}\right)^{-\alpha} \]

is an infinitely divisible characteristic function.

From this it follows, in particular, that the Laplace distribution (Exercise 6 of Chapter 5) is infinitely divisible.

Exercise 6:

The density function of a random variable \(\xi\) is
\[ p(x)=\frac{1}{2 \alpha} e^{-\frac{|x-a|}{\alpha}} \]

(the Laplace distribution). Find \(\mathbf{M}_{\xi}\) and \(\mathbf{D \xi}\).