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      All Questions

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      -1
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      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
      1answer
      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
      1answer
      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
      0answers
      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
      0answers
      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
      1answer
      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
      1answer
      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
      1answer
      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
      0answers
      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
      0answers
      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
      1answer
      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
      0answers
      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
      2answers
      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
      0answers
      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
      1answer
      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 $\eta(B_1), \ldots, \eta(B_n)$...

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