# Version of Donsker-Invariance-Principle

Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$. Define $\bar{X}_t(u):=\tilde{X}_{tu}$. Can we say, that for fixed $u\in[0,1]$

$$\frac{\bar{X}_{t}(u)}{\sqrt{t}}\xrightarrow{d} W_{u}$$ as $t \rightarrow \infty$, where $W$ is a standard Wiener process?

More generally, do we have the convergence $$\left(\frac{\bar{X}_{t}(u)}{\sqrt{t}}\right)_{u\in[0,1]}\xrightarrow{d} (W_{u})_{u\in[0,1]}\text{?}$$

## 1 Answer

Let $Y_t(u):=\frac{\bar{X}_{t}(u)}{\sqrt{t}}$ and $W(u):=W_u$. The convergence (as $t\to\infty$) in distribution (in the Skorokhod space $D[0,1]$) of $Y_t$ to $W$ can be proved quite similarly to the way it was done e.g. in the proof of Theorem 16.14 in Foundations of Modern Probability by Kallenberg.

Alternatively and a bit more directly, one may use e.g. Theorem 15.6 in Convergence of Probability Measures by Billingsley (1968). Indeed, the convergence of the finite-dimensional distributions of $Y_t$ to those of $W$ follows by the convergence of the one-dimensional distributions and the independence of the increments of the processes $Y_t$. In view of the condition $Var\;X_1=1$, condition (15.21) in Billingsley holds (with $Y_t$ in place of $X_n$) for $\gamma=2$, $\alpha=1$, and $F(u)\equiv u$ -- cf. the condition sufficient for (15.21) displayed right after Theorem 15.6 in Billingsley.

• thanks for you support. I try to reamphasize your second statement on Theorem 15.6.. Why doesn't from $(Y_t(s_1),\ldots ,Y_{t}(s_k))\xrightarrow{d} (W(s_1),\ldots,W(s_k))$ for $0\leq s_1< \ldots<s_k\leq 1$ (are times of continuity of W, which is fullfilled anyways) as $t\rightarrow \infty$ (this is assumption (15.20) in Billingsley), where $W=(W(s))_{s\leq 1}$ follow directly $(Y_t(s))_{s\leq 1}\xrightarrow{d} (W(s))_{s\leq 1}$? I mean the process is completely determined by its finite dimensional distribution. Why do we need more than (15.20) ? Best regards ziT – ziT Jul 19 '16 at 12:34
• You did not specify the meaning of the symbol $\xrightarrow{d}$. Usually, it means the convergence in distribution, in an appropriate space. In this case, concerning convergence of Levy processes, the natural space is the Skorokhod one, $D[0,1]$. Whereas a probability distribution in $D[0,1]$ is indeed uniquely determined by the finite-dimensional distributions, the questions of convergence and even the existence of a random function in $D[0,1]$ with given finite-dimensional distributions (see Theorem 15.7 in Billingsley) are quite different matters. – Iosif Pinelis Jul 19 '16 at 13:42
• [Previous comment continued:] Condition (15.21) in Billingsley is needed to establish the tightness, which is necessary for the convergence. The convergence in distribution in $D[0,1]$ is much more than that of the finite-dimensional distributions; recall the Portmanteau theorem (Theorem 2.1 in Billingsley). In particular, the convergence in distribution of $Y_t$ to $W$ implies that $P(Y_t(u)<g(u)\ \forall u\in[0,1])\to P(W(u)<g(u)\ \forall u\in[0,1])$ as $t\to\infty$, for any (say) continuous function $g$. – Iosif Pinelis Jul 19 '16 at 13:42
• Ok thanks. I prefer your 2. statement of billingsley then. Because bY the convergence of $Y_t(1)$ to $W(1)$ the convergence to the Brownian law on the skorohodspace on $[0,1]$ follows directly with your remarks, if i am right. – ziT Jul 19 '16 at 15:16