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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 Loewner local growth property gives a continuous Loewner driving function, and we normalize $U_0=0$. Mapping out an initial hull transforms the future driver to $U_{t+s}-U_t$. The Conformal Markov property of SLE therefore gives stationary increments and independent increments.

Consequently $U$ is a continuous Lévy process. In the Lévy–Khintchine formula, continuity excludes its jump measure, so its only possible components are a deterministic linear drift and Brownian motion:Conformal invariance under dilations gives $U_{r^{2}t}/r\stackrel{d}{=}{U}_t$. Comparing the expectations of these Gaussian random variables yields $\mu rt=\mu t$ for every $r>0$, hence $\mu =0$. The Brownian term already has the required law by Brownian scaling. Therefore the characterization of the SLE driving function isThe case $\kappa =0$ is the constant zero driver. No reflection-symmetry assumption is needed to remove the drift; scaling suffices.

A real Lévy process satisfies $X_0=0$ almost surely, has stationary independent increments, is stochastically continuous, and is taken in its almost surely càdlàg version.

A Lévy process starts at zero almost surely, has stationary independent increments, and is stochastically continuous; one normally takes its càdlàg modification.

A Lévy process $(X_t{)}_{t\ge 0}$ starts at zero, has independent and stationary increments, is stochastically continuous, and is taken with càdlàg sample paths. Thus for $0\le t_0<\cdots <t_n$, the increments $X_{t_j}-X_{t_{j-1}}$ are independent, and the law of $X_{s+t}-X_s$ depends only on $t$.

Let $T=\inf \{t\ge 0:X_t\notin (-a,b)\}$, and suppose the jumps of $X$ have absolute value at most $c$. A nonconstant centered finite-variance Lévy process oscillates, so $T<\infty$ almost surely. Before $T$ the process lies in $(-a,b)$, and at $T$ its bounded overshoot gives $X_T\in [-a-c,b+c]$. Thus the variables $X_{T\wedge n}^2$ are uniformly bounded.

Apply the optional sampling theorem for a supermartingale to the martingale from part (c):Bounded convergence theorem on the left and monotone convergence theorem on the right yield

For the unit-rate continuous-time symmetric simple random walk, the processes $n^{-\alpha }{X}_{nt}$ have three finite-dimensional regimes. For $0<\alpha <1/2$ no probability limit is possible, because the one-time characteristic functions tend to zero off the origin and are discontinuous there. At $\alpha =1/2$ the limit is standard Brownian motion. For $\alpha >1/2$ it is the identically zero Lévy process. This follows from $n(1-\cos (n^{-\alpha }\theta ))\sim ({\theta }^2/2)n^{1-2\alpha }$ and independent increments. Thus a nonzero limit uniquely determines the exponent, whereas merely asking for a Lévy limit does not.

If $X$ is a Lévy process and $T$ an independent subordinator, then $Y_t=X_{T_t}$ is a Lévy process. Conditional on the clock, increments of $X$ over disjoint clock intervals are independent; averaging over the independent stationary increments of the clock gives the same properties for $Y$. If $\mathbb{E}{e}^{iu{X}_s}=e^{-s\Psi (u)}$, its characteristic function is $\mathbb{E}{e}^{-T_t\Psi (u)}$. For standard Brownian motion, $\Psi (u)=u^2/2$, so this is the clock's Laplace transform at $u^2/2$.

A subordinator is a Lévy process with nondecreasing paths. Its nonnegative stationary increments have a Laplace transform of the formwhere $\Phi$ is called its Laplace exponent. Its standard path convention is càdlàg. Subordinators provide random clocks for subordination of a Lévy process.