**-3**

votes

**0**answers

56 views

### What's the probability that at least one of k top data still in top k positions of a data set with error? [on hold]

Let {${d_1,d_2,d_3,..., d_k,...d_n}$} is a descendingly sorted data set.
Now we suppose that each data in the set has a probability $p_e$ to go wrong.
The problem is what's the probability that at ...

**1**

vote

**0**answers

37 views

### When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$
...

**0**

votes

**0**answers

60 views

### Estimating the moments of a random variable

Suppose i wanted to estimate the expectation and variance of a random variable $X$. More over suppose i could write a variable $X$ as a sum of indicator random variables $X=\sum_{i=1}^{k} X_{i}$. Are ...

**2**

votes

**1**answer

46 views

### Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to).
I have a ...

**6**

votes

**5**answers

670 views

### Collisions between rooks taking random flights on an N by M chessboard

I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:
(1) We choose one of the $k$ ...

**-2**

votes

**0**answers

26 views

### Mean time for the renewal process [on hold]

The system is as below.
Energy keeps coming at a node with a constant rate $\rho$.
Node has files of size exponential($\lambda$) to be transmitted. At time zero, say the energy at the node be zero. ...

**2**

votes

**0**answers

53 views

### Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...

**2**

votes

**1**answer

44 views

### Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...

**5**

votes

**1**answer

238 views

### lower-bound for $Pr[X\geq EX]$

Given n random variables, $X_1, ..., X_n$, each takes value 0 or $a_i \in[0, 1]$. $X = \sum_{i=1}^n X_i$ and $EX \geq 1$ is the expected value of $X$. Can we get a lower-bound for $Pr[X \geq EX]$? It ...

**0**

votes

**1**answer

343 views

### Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...

**1**

vote

**0**answers

28 views

### Reference for a special case of the Hanson-Wright inequality

I would like find tail bounds for the expression
$$
\begin{align*}
\left|\left\langle a,\phi\right\rangle \left\langle \phi,b\right\rangle -\left\langle a,b\right\rangle\right|,
\end{align*}
$$
where ...

**1**

vote

**1**answer

68 views

### Perturbation of a Bessel process of dimension 2

Bessel process of dimension 2 is defined to be solution of
$$
dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0
$$
where $B$ is a standard 1-dimensional Brownian motion.
$X$ can be viewed as the norm of a ...

**8**

votes

**3**answers

265 views

### Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...

**-2**

votes

**0**answers

38 views

### Expected value of minimum of an exponential function [on hold]

Find expected value of minimum of n random variables:
x = (x1,x2,x3,..,xn)
The distribution is an exponential function:
...

**0**

votes

**0**answers

55 views

### Number of graphs with M edges that does not contain K-clique [on hold]

If we consider the space of graphs $G(n,M)$ where $M$ denotes the number of edges. Is there any known way of calculating the number of graphs within this space that does not contain any k-cliques? Can ...

**30**

votes

**1**answer

5k views

### “psi-epistemic theories” in 3 or more dimensions

In their recent paper The Quantum State Can Be Interpreted Statistically, Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for ...

**4**

votes

**1**answer

262 views

### Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by:
$X_{i}= h(i) + \varepsilon_i $,
$h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$
where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...

**2**

votes

**1**answer

204 views

### Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian?
In ...

**0**

votes

**0**answers

58 views

### Approximation of quadratic variation

Here $M$ and $N$ are two bounded continuous martingales with respect to some filtration $(\mathcal F_t)_t$. I found this claim in a paper I was reading: $t$ being fixed, then a.s.
$$\lim_{h\rightarrow ...

**2**

votes

**0**answers

64 views

### Restricted singular values of random matrix

Let $X \in \mathbb{R}^{p\times p}$ be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of $X$ follow the Marchenko-Pastur law.
Now let's ...

**4**

votes

**1**answer

402 views

### Given a Levy Exponent find the jump-measure and drift

A Levy subordinator is an finite variation Levy process with non-negative drift and positive jumps. The Levy exponent is given by
$$\phi(\lambda) = \gamma \lambda + \int_0^\infty ( 1 - e^{-\lambda ...

**1**

vote

**1**answer

205 views

### MMSE estimator expressed through cumulants

I have a linear model $$Y=HX+N,$$ where $H$ is a matrix and $X$ are drawn from $p_X(X)$, and $N$ is Gaussian noise variates.
Now, if $X$ is multivariate Gaussian, then a linear estimator ...

**1**

vote

**2**answers

170 views

### A special class of random variables

I'm interested in classes C of $R^1$-valued random variables which possess the following properties:
1) the sum of two independent random variables from class C belongs to class C;
2) for any ...

**1**

vote

**1**answer

162 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

**0**

votes

**0**answers

27 views

### Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...

**-8**

votes

**0**answers

66 views

### A true-false exam has five questions. Andy is completely ignorant and so he tosses a fair coin to answer each question [on hold]

A true-false exam has five questions.
Andy is completely ignorant and so he tosses a fair coin to decide his answer to each question.
What is the probability that he scores at least four correct?

**1**

vote

**0**answers

48 views

### Asymptotic variance for partial sum of a stationary process

Let $X = (X_1, \dots, X_n, \dots)$ be a sequence of random variables. We assume that the process X is stationary i.e. for any integer $k$, any set of indices $i_1 < \dots < i_k$ and any integer ...

**5**

votes

**1**answer

252 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**0**

votes

**1**answer

90 views

### Help in finding distribution of the following function of random variable [on hold]

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$
$$X_2 \sim \mathcal{CN}(0,\sigma)$$
If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...

**0**

votes

**1**answer

205 views

### On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
...

**0**

votes

**1**answer

111 views

### Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...

**4**

votes

**1**answer

281 views

### Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...

**1**

vote

**0**answers

54 views

### Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem:
Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$.
Let $a( x;. )$ and $f(x;.)$ be ...

**3**

votes

**0**answers

116 views

### Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback.
Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...

**-3**

votes

**3**answers

278 views

### Determinant of matrix from set {-1, 1} [on hold]

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...

**0**

votes

**1**answer

248 views

### About an integral equation

I would like to obtain $g$ by solving the following integral equation
$$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$
where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+
...

**5**

votes

**1**answer

252 views

### Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...

**2**

votes

**1**answer

148 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**7**

votes

**1**answer

399 views

### Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
...

**5**

votes

**1**answer

259 views

### How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...

**12**

votes

**4**answers

2k views

### Correlated Brownian motion and Poisson process

Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ?
I first asked this question ...

**1**

vote

**0**answers

84 views

### First passage time of a pure drift process

I am facing the following unusual problem: $Z_t$ is a pure drift process of the form
$$ dZ_t = \kappa(X_t - Z_t) dt $$
where $X_t$ is another bounded process.
I am interested in computing / ...

**0**

votes

**0**answers

35 views

### What is the Vapnik-Chervonenkis dimension of sigmoidal functions? [migrated]

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...

**3**

votes

**2**answers

211 views

### Expectation of a generalization of Dirichlet distribution

For the standard Dirichlet, the expectation of $X_i$ is $\alpha_i/\alpha_0$, where $\alpha_0 = \sum_i \alpha_i$ [http://en.wikipedia.org/wiki/Dirichlet_distribution].
I am considering the following ...

**4**

votes

**1**answer

172 views

### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...

**22**

votes

**6**answers

1k views

### Shortest grid-graph paths with random diagonal shortcuts

Suppose you have a network of edges connecting
each integer lattice point
in the 2D square grid $[0,n]^2$
to each of its (at most) four neighbors, {N,S,E,W}.
Within each of the $n^2$ unit cells of ...

**5**

votes

**2**answers

287 views

### Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$.
Our goal is to find $t\in[0,1]$ such that
$$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$
Is there a closed-form ...

**0**

votes

**1**answer

383 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**0**

votes

**1**answer

55 views

### Probability spaces involved in using Bayesian Inference

I am currently reading "Statistical and Inductive Inference by Minimum Message Length" by C.S. Wallace. In this, Wallace gives a fairly informal account of Bayesian Inference which, in the case ...

**0**

votes

**1**answer

59 views

### Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary?

Consider the following function class: $F={f:R^d\rightarrow [a,b], f(x)=\sigma(w^Tx)}$ where $\sigma(.)$ is Lipschitz, and $w\in R^d$ is a parameter vector. The problem I'm working on is a machine ...