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The printed uniqueness claim omits nondegeneracy of the limit. The zero process is a Lévy process, and every $\alpha >1/2$ yields that limit. We first classify all possibilities under the stated assumptions, then identify the unique nonzero one.

For $t>0$, part (b) givesfor each fixed $\theta \ne 0$. If $\alpha <1/2$, the characteristic functions tend to zero for every nonzero parameter and remain one at zero. This discontinuous function cannot be the characteristic function of a probability law. The characteristic-function convergence theorem therefore rules out even one-dimensional weak convergence of random variables at a positive time.

For joint convergence, take $0=t_0<t_1<\cdots <t_m$ and write $q_k={\sum }_{j=k}^m{\theta }_j$. Independent increments giveAt $\alpha =1/2$, this tends to $\exp \{-\frac{1}{2}{\sum }_k(t_k-t_{k-1})q_k^2\}$. These are exactly the joint laws of standard Brownian motion: increments are independent centred normals with variances $t_k-t_{k-1}$. The assumed limiting Lévy process thus has the Brownian law.

For $\alpha >1/2$, the same joint characteristic functions tend to one. Every limiting vector is zero. The càdlàg path convention then makes the limiting process identically zero almost surely: first intersect the zero-value events at rational times, then use right continuity. The complete scaling classification of a continuous-time symmetric simple random walk isIn particular, if a nonzero limiting Lévy process is intended, the unique exponent is $\alpha =1/2$, andThe finite-dimensional Brownian law, together with the Lévy càdlàg convention, determines the process law and its continuous version. No assertion of path-space convergence of the approximating walks is needed here.

The characteristic function of $S_n/n$ isThus every $S_n/n$ already has the standard Cauchy law, and convergence in law follows from characteristic-function convergence theorem.

Characteristic-function convergence theorem says $X_n\Rightarrow X$ iff ${\Phi }_{X_n}(t)\to {\Phi }_X(t)$ for all $t$, where the limiting function is continuous at zero.