Tagged Questions

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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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Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
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310 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
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I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. [closed]

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$. means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
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Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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distribution of Brownian Motion involving integral

What is the distribution of $\int_{t}^{T} W(s)ds$? Given that W(t) is brownian motion. So far, I have the following, $\int_{t}^{T} W(s)ds$ = $(T-t)W(t) + \int_{t}^{T} (T-s)dW(s)$ Also, ...
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Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
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Variable t times a Wiener Process W(1/t)

If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that Since $W(1/t)\space \sim N(0,1/t)$ that $V(t) \sim t\cdot N(0,1/t)$? And if so then is \$t\cdot N(0,1/t) = ...