> [!definition] > > Let $\alpha > 0$ and $\mu$ be a [[Probability|probability]] measure on $\real^d$. $\mu$ is $\alpha$-stable if > 1. Its [[Characteristic Function|characteristic function]] admits a [[Principal Logarithm|principal logarithm]]. > 2. For any $t > 0$, $\ell_{\wh \mu}(t\xi) = t^\alpha \ell_{\wh \mu}(\xi)$ (equivalently $t^{-\alpha}\ell_{\wh \mu}(t\xi) = \ell_{\wh \mu}$). > [!theorem] > > Let $\mu$ be an $\alpha$-stable distribution, and $X$ be a random variable such that $\mu_X = \mu$. For any $c \in \real$, let $\nu_c$ be the distribution of $cX$, then $\nu_c$ is also $\alpha$-stable. > > *Proof*. Let $\xi \in \real^d$, then > $ > \wh \mu(\xi) = \wh \nu_c(c\xi) \quad \ell_{\wh \mu}(\xi) = \ell_{\wh \nu_c}(c\xi) > $ > so for any $t > 0$, > $ > \ell_{\wh \nu_c}(t\xi) = \ell_{\wh \nu_c}(ct\xi) = t^\alpha \ell_{\wh \mu}(c\xi) = t^\alpha \ell_{\wh \nu_c}(\xi) > $ > [!theorem] > > $\alpha$-stable distributions are [[Infinitely Divisible Distribution|infinitely divisible]]. > [!theorem] > > Let $\mu$ be an infinitely divisible distribution with canonical representation > $ > \wh \mu = e^{\ell_{m, C, M}} > $ > If $\mu$ is $\alpha$-stable, then $t^{-\alpha}\ell_{m, C, M}(t\xi) = \ell_{m, C, M}(\xi)$. Equivalently, > $ > \ell_{m, C, M}(t^{1/\alpha}\xi) = t\ell_{m, C, M}(\xi) = \ell_{tm, tC, tM}(\xi) > $ > Let $\mu_t$ be a probability measure such that $\wh \mu_t = e^{\ell_{tm, tC, tM}}$. Therefore $\mu$ is $\alpha$-stable if and only if $\mu_t$ is equal to $\mu$ under the rescaling $x \to t^{1/\alpha}x$. > [!theorem] > > 1. There is no non-$\delta$, $\alpha$-stable laws for $\alpha > 2$. > 2. The only $2$-stable laws are centred Gaussians. > 3. If $\mu$ is an $\alpha$-stable law with $\alpha \in (0, 2)$, then $C_\mu = 0$ in its canonical representation. > > *Proof*. Suppose that $\mu$ is $\alpha$-stable. Let $\xi \in \real^d$ and $t > 0$, then $\ell_{\wh \mu}(\xi)$ can be written as > $ > \frac{\ell_{\wh \mu}(t\xi)}{t^\alpha} = \frac{i\angles{m, t\xi} - t^2(\xi C \xi)/2 + \int_{\real^d}\paren{e^{i\angles{x, \xi}} - 1 - i\angles{x, \xi} \cdot \one_{B(0, 1)}} dM(x)}{t^\alpha} > $ > If $\alpha > 0$, then as $t \to \infty$, $\ell_{\wh \mu}(\xi) = 0$ and $\mu = \delta$. If $\alpha = 2$, then $\ell_{\wh \mu}(\xi) = -(\xi C \xi)/2$ and $\mu$ is a centred Gaussian. If $\alpha < 2$, then > $ > (\xi C \xi) = -2\limv{t}\frac{\ell_{\wh \mu}(t\xi)}{t^2} = -2\limv{t}\frac{\ell_{\wh \mu}(\xi)}{t^{2 - \alpha}} = 0 > $