Tagged Questions

0
votes
1answer
25 views

Examples of convergence of random variables

First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution: $X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
1
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1answer
28 views

Is $\{ r \mapsto X_{r} \text{ is continuous for all } s < t \} \in \sigma(X_s : s \leq t)$?

If $(X_t)_{t \geq 0}$ is a stochastic process, is $\{ r \mapsto X_{r} \text{ is continuous for all } $s < t$ \} \in \sigma(X_s : s \leq t)$? I'm particularly interested in the case where $X_t$ is ...
2
votes
2answers
84 views

First jump time of Poisson process (and general right-continuous processes).

I've read that the first jump time of the Poisson process is totally inaccessible (definition at the bottom for anyone interested). This made me wonder if the first jump time is a stopping time. I ...
1
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0answers
39 views

Showing a certain process has $\limsup X_t$ bounded almost surely.

This question has been solved. I'm working on a problem where I need to show $$\limsup_{t \rightarrow \infty} X_t \leq \sqrt{c}\quad \text{a.s.}$$ where $X_t$, $t \geq 0$ is a stochastic process ...
1
vote
1answer
44 views

Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
3
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0answers
60 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $ is a random variable? The kronecker product ...
0
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0answers
163 views

Random Variable on a Sphere

Not sure where to start with this problem: For any $d\geq 1$, we admit that there is only one probability measure $\mu$ on $\mathcal S_d$, (the $(d-1)-th$ dimensional sphere embedded in $\mathbb ...
0
votes
1answer
52 views

correlation between two different variables

I am studying stochastic processes and found the next problem: Let $A$ and $\Phi $ be two independent random variables such that $E(A) = 0$, $E(A^2) < \infty$, and $\Phi$ is uniformly distributed ...
1
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1answer
149 views

What is some books at the level which including this inequality and its proof?

I always wanting to looking into harder random variable/probability/stochastic process/statistics books that are harder than the intro one and have multiple random variable but easy enough to have ...
3
votes
1answer
78 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
1
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1answer
46 views

How can I show that $z_i =\cos(iw)$ where $w$ is uniform on $[0,2\pi]$ is a white noise process?

How can I show that $z_i =\cos(iw)$, where $w$ is uniform on $[0,2\pi]$ is a white noise process? So far, I have shown $E(z_i)=0$ by integrating. However, I need to show ...
1
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0answers
95 views

Independent Exponentially Distributed Random Variables - Athletes Problem??

Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...
1
vote
2answers
70 views

Independence of Brownian motion

While by definition the increments of a Brownian motion are independent, it is unclear to me whether (that implies that) the random variables $W_t$ and $W_s$ are independent for $t \neq s$. While ...
1
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1answer
121 views

Programming a set of binary switches where the lifetime of a given state is exponentially distributed

Imagine I have $(s_1, ..., s_N) \in S$ binary switches in a panel that can be switched between states $0$ and $1$. Initially, we flip all of the switches to the $0$ state. Now, for each of $(t_1, ...