# Tagged Questions

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### (Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
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### convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
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I have got the following question. Let $(S_t)_{t\in[0,T] }$ be a geometric Browninan motion. Consider a sequence of bounded random variables $(\tau_n)_{n\in\mathbb N}$ such that $\tau_n\downarrow ... 1answer 26 views ### convergence of Ito integral Suppose there is a deterministic process$\phi$in$L^2(R)$. Need to prove that$\int_0^n \phi_u dW_u$converges in$L^2(P)$to some$X\in L^2(P)$as$n\rightarrow\infty$. Also need to show that ... 1answer 97 views ### On the quadratic variation I understand that the Quadratic Variation of Brownian Motion$B_t$is$[B_t,B_t]=t$and I know that the equality is under the meaning of$\mathcal{L}^2$convergence. Yet I saw in some book saying that ... 1answer 144 views ### Laplace Transform of a Brownian motion If$v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$is a Standard Brownian motion, then for what values of$s,\omega$does the Laplace transform$l(\omega,s) = \int_0^\infty e^{-st} v(\omega,t) ...
Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
How can we show that this martingale $$e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.