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