Central Limit Theorem for Lévy Process

Question

I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_t$ without mentioning the exact theorem. Due to the infinite divisible property I can write $X_t$ as a sum of $N$ iid random variables $X^i$ $$ X_t=\sum_{i=1}^N X^i_{t/N} $$ The Problem is, that i want $t\rightarrow \infty$ but for the CLT i have to keep my sequence of my equidistant iid random variables fixed (like $t/N$ fixed). But they do change, as $t\rightarrow \infty$. The book now just says, that with the central limit theorem for Lévy Processes it holds for $t\rightarrow \infty$ \begin{align} \frac{X_t-\overbrace{tE[X_{1}]}^{=E[X_t]}}{\sqrt{t}}\rightarrow \mathcal{N}(0,\operatorname{Var}[X_1])\\ \sqrt{t} \left(\frac{X_t}{t}-E[X_1])\rightarrow \mathcal{N}(0,\operatorname{Var}[X_1]\right) \end{align} I can't find any proofs, lectures or literature about it. Can you help me out?

Answer 1

Without any additional assumptions on the Lévy process $(X_t)_{t \geq 0}$, a central limit theorem does not hold true.

Let $(X_t)_{t \geq 0}$ be a (one-dimensional) Lévy process with Lévy triplet $(b,\sigma^2,\nu)$. Define

$$T(x) := \nu((x,\infty)) + \nu((-\infty,-x))$$

and

$$U(x) := \sigma^2+2 \int_0^x y T(y) \, dy$$

for $x>0$. There is the following statement by Doney and Maller:

1. Suppose that $T(x)>0$ for all $x>0$. Then there exist deterministic functions $a(t),b(t)>0$ such that $$\frac{X_t-a(t)}{b(t)} \stackrel{t \to \infty}{\to} N(0,1) \tag{1}$$ if, and only if, $$\frac{U(x)}{x^2 T(x)} \stackrel{x \to \infty}{\to} \infty.$$
2. Suppose that $T(x)=0$ for all $x>0$ (i.e. the Lévy measure $\nu$ is symmetric). Then $(1)$ holds if, and only if, $\sigma^2>0$. In this case, $a(t) = t \mathbb{E}(X_1)$ and $b(t) = \sigma \sqrt{t}$ is admissible.

In dimension $d>1$ there are CLT-results for Lévy processes by Grabchak.

References:

• R.A. Doney and R.A. Maller.: Stability and Attraction to Normality for Lévy processes at Zero and at Infinity. Journal of Theoretical Probability, Vol. 15, July 2002.
• Michael Grabchack: A note on the multivariate CLT and convergence of Lévy processes at Long and Short Times. ArXiV.