= Solution
Disjoint time strips of the <Poisson random measure> are <independent>, and its intensity is translation invariant in time. The construction and its limits therefore give <independent increments> and <stationary increments>, with
$$
\mathbb E e^{iu(X_t-X_s)}=e^{-(t-s)|u|}\qquad(0\leq s<t).
$$
For the scaling actually printed in the original PDF,
$$
\mathbb E e^{iu\alpha X_{\alpha t}}=e^{-\alpha^2t|u|}.
$$
For example, $\alpha=2$, $t=1$, $u=1$ gives $e^{-4}$ rather than $e^{-1}$. Thus \b[the requested equality of process laws is false unless $\alpha=1$].
The intended <self-similarity of a Cauchy process> rescales space and time in opposite directions:
$$
\boxed{\left(\alpha^{-1}X_{\alpha t}\right)_{t\geq0}\stackrel{d}=(X_t)_{t\geq0}}
\qquad\text{or equivalently}\qquad
\boxed{\left(\alpha X_{t/\alpha}\right)_{t\geq0}\stackrel{d}=(X_t)_{t\geq0}}.
$$
Indeed, for $0\leq s<t$,
$$
\mathbb E\exp\left[iu\alpha^{-1}(X_{\alpha t}-X_{\alpha s})\right]
=\exp\left[-\alpha(t-s)|u|/\alpha\right]=e^{-(t-s)|u|}.
$$
The transformed process retains <independent increments>. For any ordered times, the joint <characteristic function> of its successive increments is therefore the same product as for $X$; cumulative summation proves equality of all <finite-dimensional distributions>. Both constructions have <càdlàg> paths, so these distributions also determine their laws on the usual <path space>. This derives the corrected invariance and explicitly diagnoses the false printed assertion.
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