= Solution
\b[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) gives
$$
\mathbb E e^{i\theta X_t^{(n)}}
=\exp\{-nt(1-\cos(n^{-\alpha}\theta))\},
\qquad
n(1-\cos(n^{-\alpha}\theta))
=\frac{\theta^2}{2}n^{1-2\alpha}(1+o(1))
$$
for each fixed $\theta\ne0$. 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> give
$$
\mathbb E\exp\left(i\sum_{j=1}^m\theta_jX_{t_j}^{(n)}\right)
=\exp\left\{-n\sum_{k=1}^m(t_k-t_{k-1})\left(1-\cos(n^{-\alpha}q_k)\right)\right\}.
$$
At $\alpha=1/2$, this tends to $\exp\{-\frac12\sum_k(t_k-t_{k-1})q_k^2\}$. 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> is
$$
\boxed{\begin{array}{c|c}
0<\alpha<1/2&\text{no finite-dimensional probability limit}\\
\alpha=1/2&Y\text{ is standard Brownian motion}\\
1/2<\alpha\leq1&Y_t\equiv0.
\end{array}}
$$
In particular, \b[if a nonzero limiting Lévy process is intended, the unique exponent is $\alpha=1/2$], and
$$
\boxed{Y_t\sim\mathcal N(0,t),\qquad \mathbb E[Y_sY_t]=\min(s,t).}
$$
The 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.
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