Self-similarity and stability

Let $X$ be stable, and let $X_{1}, X_{2}, \ldots , X_{n}$ be a random sample distributed as $X$. Show that, for $n \geq 1$, there exist $A_{n}, B_{n}$ such that $X_{1}+X_{2}+\cdots +X_{n} \stackrel{\mathrm{D}}{ = } B_{n} X+A_{n}$.

Prove that, for a non-degenerate, self-similar process $X$ with stationary increments, finite variances, and Hurst exponent $H > 0$,

Show that a stable distribution is infinitely divisible.

Show that a self-similar Lévy process either is a Wiener process or has infinite variances.

Prove that a distribution with characteristic function $\phi (\theta ) = \exp \left(-|\theta |^{\alpha }\right)$, where $\alpha \in (0,2]$, is strictly $\alpha$-stable and symmetric.

Let $X$ and $Y$ be independent stable random variables, where $X$ is symmetric with exponent $\alpha \in (0,2]$, and $Y$ is non-negative with Laplace transform $M(\theta ) = \exp \left(-k \theta^{\beta }\right)$ for $\theta \in [0, \infty )$, where $1 \neq \beta \in (0,2]$. Show that $Z = X Y^{1 / \alpha }$ is symmetric and $\alpha \beta$-stable. Deduce that, for independent $N(0,1)$ random variables $U, V$, the ratio $Z = U / V$ has the Cauchy distribution.