# All Questions

Tagged with stochastic-processes pr.probability

934
questions

**-1**

votes

**0**answers

21 views

### Distribution of markov chain with a stopping time

I have a Markov chain $X_0, X_1, ..., X_\tau$, where $X_0$ is sampled from the stationary distribution and $\tau$ is a stopping time. Is it true that for any fixed $k$, $X_k$, given that $k \leq \tau$,...

**1**

vote

**1**answer

40 views

### Comparing noisy truncated RV with noisy regular RV

For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...

**1**

vote

**1**answer

67 views

### Ito's Formula for functions that are $C^2$ a.e

In the conventional Ito's formula, it is required that $F$ is $C^2$ everywhere. However I've seen mentioning of a slightly weaker condition, where $F$ is $C^1$ everywhere but $C^2$ a.e.. Is there any ...

**3**

votes

**0**answers

60 views

### Beta distribution and Wiener process

Suppose $W(t)$ is the standard Wiener process on $[0; 1]$ and $\{T_x\}_{x \in \mathbb{R}}$ is a collection of random variables defined by the following relation:
$$T_x = \mu(\{t \in [0;1] | W(t) > ...

**0**

votes

**0**answers

66 views

### Conditioned Brownian motion?

Let $U\subseteq (C_0[0,1];\mathbb{R})$ be an open subset of the Wiener space satisfying $0<\gamma(U)<1$; where $\gamma$ is the Wiener measure and let $W_t$ be the standard (Wiener) coordinate ...

**-1**

votes

**1**answer

43 views

### Convergence in mean and convergence in distribution

Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that
$$
0< ...

**1**

vote

**1**answer

66 views

### About another potential characterization of normal numbers

Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and ...

**0**

votes

**1**answer

97 views

### Convergence in distribution of products

Suppose that a sequence of random variables $Y_n$ convergence in $L^2$ to $Y$, i.e.
$$
E|Y_n-Y|^2\to0\quad \text{as}\quad n\to\infty.
$$
Moreover, there exist constants $c_0$ and $c_1$ such that
$$
0 &...

**3**

votes

**0**answers

102 views

### Reference: hitting time of Gaussian process

Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by
$$
Y_t = y+\int_0^t X_s ds + W_t,
$$
for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...

**1**

vote

**0**answers

54 views

### Approximation of measured-valued function by continuous functions

For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e.,
$$
\int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty.
$$
Let $\mu$ be a probability measure on $R^d$ such that
$$
\int_{R^d}\int_{R^d}(|z|^2\...

**-1**

votes

**1**answer

85 views

### Approximation of function in general measure space

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...

**0**

votes

**0**answers

49 views

### Concentration of Sample Mode

Is there a concentration bound for sample mode, when there exists a unique mode for a density $f$ that doesn't depend on $f$ (Essentially I am looking for a Chernoff type bound for mode)?.

**2**

votes

**2**answers

178 views

### Probability space with exactly one Brownian motion

Very recently, the following question was asked:
Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. ...

**1**

vote

**0**answers

64 views

### Wiener isometry for semimartingales

Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying
$$
\mathbb{E}\left[
\int_0^{\...

**5**

votes

**1**answer

80 views

### De Finetti-style theorem for Point Processes

I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies:
Complete independence (the random variables $\eta(B_1), \ldots, \eta(B_n)$...