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Let $(Y_t)_{t \in [0,1]}$ be a Lévy process and denote by $\Delta Y_t := Y_t-Y_{t-}$ the jump height at time $t$. The aim is to show that $(Y_t)_{t \in [0,1]}$ has only finitely many jumps with jump size larger than $r$, i.e. that $$\{t \in [0,1]; |\Delta Y_t(\omega)|>r\}$$ is a finite set for (almost) all $\omega \in \Omega$. This follows from the ...


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Every Polish space without isolated points admits an atomless strictly positive Borel probability measure. This is a consequence of the fact that such a space has a dense subspace homeomorphic to the irrational reals. The latter is mentioned at the end of exercise 6.2.A in Engelking's General topology.


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Let $B_N = \bigcup_{n \ge N} A_n$. Then $A_n \ \text{i.o.} = \bigcap_{N=1}^\infty B_N$. Note that $B_{N+1} \subseteq B_N$. Then countable additivity implies $P(\bigcap_{N=1}^\infty B_N) = \lim_{N \to \infty} P(B_N)$.


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Let $\left(\Omega,\mathcal{A},P\right)$ be a probability space. Then a random variable on it is a function $X:\Omega\rightarrow\mathbb{R}$ such that $X^{-1}\left(B\right)=\left\{ \omega\in\Omega\mid X\left(\omega\right)\in B\right\} \in\mathcal{A}$ for each Borelset $B$. Denoting the collection of Borelsets on $\mathbb{R}$ by $\mathcal{B}$ we state that ...


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Your proof is correct. More generally: If $E(X_n)\to C$ and $\mathrm{var}(X_n)\leqslant\epsilon_n$ where the series $\sum\limits_n\epsilon_n$ converges, then $X_n\to C$ almost surely. Hint: There exists some positive sequence $(\alpha_n)$ such that $\alpha_n\to0$ and $\sum\limits_n\frac{\epsilon_n}{\alpha_n^2}$ converges. Consider the events ...



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