# Tag Info

1 vote

• 13.5k
Accepted

### A Gaussian process and a Rademacher proecss are sub-Gaussian

Gaussian Process. Let $Z=(Z_1,\cdots ,Z_d), t=(t_1,\cdots ,t_d), s=(s_1,\cdots, s_d)$. By definition, $Z_i \sim N(0,1)$ are independent Gaussian variables (Jointly normal random variables that are ...
• 1,451
1 vote

### What to call a sequence of Bernoulli trials with different probabilities?

It is Poisson Binomial Distribution : In Probability theory and Statistics, the Poisson Binomial Distribution is the Discrete Probability Distribution of a sum of independent Bernoulli trials that are ...
• 9,809
Accepted

### Sub-Gaussian $X_t$, prove $\mathbb{E}\left[\sup_{t\in T}X_t \right] \leq 2 \mathbb{E}\left[\sup_{\rho(t,s)\leq \delta}(X_t-X_s) \right]+J(\delta,T)$

For this problem, bounding the expectation of the supremum is crucial. The following inequality can help in this case. Proposition. Let $\{Z_i\}_{i=1}^{N}$ be $\sigma^2$-sub-Gaussian random variables. ...
• 1,451

### Help with Integration by Parts for a Markov Chain

OP here, I think I was able to figure out the problem. Part 1: Let's start with the Q Matrix: $$Q = \begin{bmatrix} B & A=-BI \\ 0 & 0 \end{bmatrix}$$ Within this matrix, we are interested ...
• 3,180
1 vote

### What to call a sequence of Bernoulli trials with different probabilities?

If $p$ is fixed and all the experiments are Bernoulli trials then we can call it a Binomial Distribution. If $p$ is not fixed, then it is more useful to consider the Bernoulli trials as separate ...
• 1,507
1 vote

### Transition probability density function for a non-trivial diffusion process.

Here we will only focus on finding the eigenfunctions of the infinitesimal generator. In other words we want to solve the following ODE below: \left(\mu z^{\beta_1} \frac{d}{d z} + \...
• 11.4k
1 vote

### Solve SDE $dX_t = X_tW_tdt + dW_t,$

To solve problems like these, rather than trying to guess the correct process to introduce, it may be easier to just let $Y_t := e^{Z_t}X_t$ where $dZ_t = \alpha_t dt + \beta_t dW_t$ for some ...
• 13.5k

• 25.8k
Accepted

• 24k

### Creating a martingale given

You are correct, the filtration by construction makes the process adapted. You also need to check the integrability condition of the definition. To check the martingale property, you need to use the ...
• 866
1 vote

### Properties of a transient state in a Markov Chain

As you have pointed out, $\sum\limits_{n=1}^{\infty}P_{jj}^{n}<\infty$, and that $\mathbb{P}_{ij}(s)=\mathbb{F}_{ij}(s)\mathbb{P}_{jj}(s)$, what you need follows directly from Abel's theorem, since ...
1 vote
Accepted

• 24k

### The almost sure event in the law of the iterated logarithm for the Brownian motion: what it looks like

The event you have captured in more explicit terms is not $\{\limsup_{t\to 0}W_t/h(t)=1\}$, but rather $\{\limsup_{t\to 0}|W_t/h(t)-1|=0\}$. The placement of the absolute value is crucial!
• 25.8k

• 12.8k

### Convergence Rate of "Infinite Monkey"-type-probability

I found a solution to the problem that is sufficient for my use case. This is not the exact convergence rate. If somebody still manages to find it in time I'll still give the bounty of course. However,...
• 107
1 vote

### Using Ito's Lemma to take a stochastic integral

Please avoid to ask too many questions at the same time in future posts. Let's tackle the side questions first. First side question. I mean, why not ? If you have a stochastic variable $X_t$ itself ...
• 8,318
Accepted

### How can I show that the first exit time by a planar Brownian motion is a.s. finite, i.e. $\mathbb{P}_z(\tau_D<\infty)=1$?

My first question is, where do i need that $W_t$ is a complex Brownian motion, I mean why can't I only work with $B_t$ instead of $W$? You could do it with just using neighborhood-recurrence for 2d-...
• 3,611
Accepted

### The distribution of the first hitting time for the Constant Elasticity of Variance process.

We are going to prove that the Laplace transform $(3)$ approaches the correct limit when $\beta \rightarrow 1_+$. We denote ${\mathfrak N}:= 1/(2(-1+\beta)$ and $\theta := 2 \mu/\sigma^2$ and we have: ...
• 11.4k

### Markov Property of a Ito Process

Rather than bringing in $B_{t+h}-B_h$, consider using $B_{t+h}-B_t$: Because $X^x_{t+h} = X^x_t\cdot\exp(ch+\alpha(B_{t+h}-B_t))$, and $B_{t+h}-B_t$ is independent of $\mathcal F_t$ with the same ...
• 25.8k

### Talagrand's functional and Dudley's sum (Vershynin 8.5.2)

I'm writing with respect to the recent version of Vershynin, i.e. it is $1+\log k$ instead of $1+\log n$. Choosing $T_k$ to be the first $2^{2^k}$ vectors will not work as suggested by the OP (or as ...
• 1,165
1 vote
Accepted

### How to prove $\sup_{t\geq 0} \mathbb{E}(M_t^2)<\infty$ implies $\mathbb{E}(\sup_{t\geq 0}M_t^2)<\infty$?

Let $f_{N}:=\sup_{N\geq t\geq 0}M_{t}^{2}$, then by (liminf)-Fatou's lemma we get $$E[\liminf_{N}f_{N}]\leq \liminf_{N}E[f_{N}]\leq 4\liminf_{N}E[M_{N}^{2}].$$ Since the sequence $f_{N}$ is increasing,...
• 3,611