**-1**

votes

**0**answers

34 views

### Distribution of $e^*f$, if $e$ is a complex Gaussian vector and $f$ is a unit norm complex vector

Let $e$ be a complex Gaussian vector where its elements are of zero mean and variance equals to $\sigma^2$. In addition, we define $f$ as a complex random unit norm vector uniformly distributed. Note ...

**1**

vote

**1**answer

84 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot ...

**6**

votes

**1**answer

107 views

### Inequality for the maximum of Gaussian variables

Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors
with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix
$\Gamma_Y-\Gamma_X$ is positive definite. Is it ...

**-2**

votes

**0**answers

76 views

### exponential tail bound for conditional probability [on hold]

I am aware of exponential tail probabilities for unconditional probability (for ex: Normal). Are there any similar results available for conditional probability (w.r.t to a sigma field) in literature ...

**1**

vote

**1**answer

82 views

### Tight binomial left tail bound

Let $X \sim \text{Bin}(n,p)$. Wikipedia claims
$$\mathbf P[X \leq (p-\epsilon)n ] \leq e^{ - 2 \epsilon^2 n}.$$
This follows from Hoeffding's inequality ...

**7**

votes

**1**answer

119 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

**-1**

votes

**0**answers

67 views

### A statistical conundrum [on hold]

I have a statistical problem in a domain that I can not talk about due to a NDA. However, I have worked out how to describe an exactly analogous problem as follows.
You are in charge of record ...

**2**

votes

**2**answers

132 views

### Bound that random walk stays within with constant probability?

For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$.
What is the bound, in terms of $n$, such that if the ...

**0**

votes

**0**answers

54 views

### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...

**0**

votes

**0**answers

63 views

### Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...

**1**

vote

**0**answers

89 views

### probability of a quadratic form being nonnegative at a random point

I am looking for good and explicit lower and upper bounds on the probability that $x^TGx\ge 0$, where $G$ is a symmetric matrix with zero trace and $x$ is a vector whose components are independent ...

**3**

votes

**0**answers

59 views

### Lower semi-continuity of the Hellinger-Fisher-Rao distance

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
...

**1**

vote

**0**answers

99 views

+50

### A path optimisation problem

Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...

**-1**

votes

**0**answers

33 views

### weak convergence related problem [closed]

I am studying for an exam and this was one of the hard problems in my textbook.
Let $f_n(x) = 1 −\cos(2\pi nx)$ for n $\in$ N and x $\in$ $[0,1]$. Verify that $f_n$ is the density of a probability ...

**0**

votes

**0**answers

41 views

### CLT related problems [closed]

Let $S_n$ = $X_1+···+X_n$ be a sum of i.i.d. mean zero random variables with a finite, positive variance. How can I show that $lim_{n\rightarrow\infty}$ $S_n/\sqrt n$ = $\infty$ a.s. also there is no ...

**1**

vote

**0**answers

53 views

### Chain Rule for Maximal Correlation

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**1**

vote

**0**answers

33 views

### An inequality for Maximal Correlation over a Markov Chain

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**0**

votes

**0**answers

64 views

### Example that a finite collection of contractions can't approach the set of all contractions well enough

I'm looking for an example of a seperable and complete metric space $(S,d)$ such that there exist some $\varepsilon > 0$ and $P$ a probability measure on the Borel open sets $\mathcal{B}_S$ for ...

**0**

votes

**0**answers

26 views

### positive Harris recurrent, aperiodic, stationary Markov chain

How to proof that every positive Harris recurrent, aperiodic, stationary Markov chain is alpha-mixing (strong-mixing)?

**1**

vote

**1**answer

90 views

### Classification of Lebesgue-Rokhlin spaces

I am currently trying to grasp some ideas on Lebesgue-Rokhlin spaces from Bogachev, "Measure Theory", vol. 2.
Such spaces are also known as standard probability spaces but the definitions are not ...

**2**

votes

**1**answer

38 views

### Edge-perspective degree distribution

I was reading this paper when I came across something called the edge-perspective degree distribution in a network. Consider a graph $G$, the degree distribution of whose nodes is $f(d)$. They say the ...

**3**

votes

**0**answers

114 views

### Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers:
(1) On the probability of satisfying a ...

**1**

vote

**1**answer

62 views

### Differentiability of stochastic process

Is it possible to construct a stochastic process $X_t$ where the limit
$\lim_{\Delta \rightarrow 0} \rm{Var}\left(\frac{X_{t_0+\Delta}-X_{t_0}}{\Delta}\right)$
does not exist but the sample paths ...

**-4**

votes

**0**answers

23 views

### Computing preference using mean and variance? [closed]

I have 10 Items, which are available for me to interact with. Interaction with them yields me some reward.I have an expectation/mean/probability and variance/uncertainty of the reward for each item. ...

**3**

votes

**0**answers

99 views

### The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...

**-1**

votes

**0**answers

64 views

### Probability distribution of the distance between two random, uniformly distributed points in the unit ball [closed]

Take two vectors, $p1$ and $p2$, that are uniformly distributed within the unit ball. Let $d = \|p1-p2\|$. What probability distribution do we have for $\frac{1}{d^2}$?
Let us consider the ...

**0**

votes

**0**answers

58 views

### Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
...

**-1**

votes

**0**answers

76 views

### What the number of the components of a specific subgraph? [closed]

Given a n vertices graph $G$, take two edge-disjoint matchings in $G$, namely $M_{1}$ and $M_{2}$, such that they cover $n-\alpha$ vertices each. In our case, $\alpha$ can be a constant or a function ...

**7**

votes

**1**answer

216 views

### Base schemes and Bayesian priors

One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme.
In Bayesian ...

**1**

vote

**1**answer

100 views

### Independence of two random variable

Let $W$ and $S$ are two positive valued continuous random variable. Suppose
$g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ ...

**11**

votes

**2**answers

365 views

### How many random sieve operations to decimate the set {2,…,n}?

Let $S$ be the set of integers $\{2,3,4,\ldots,n\}$.
Consider the following process:
Select a random element $k \in S$.
Remove from $S$ every number divisible by $k$.
Repeat with this reduced $S$.
...

**2**

votes

**1**answer

35 views

### Conditioned binomial dominates unconditioned with different parameter

Let $X \sim \text{Bin}(n,p)$ and $Y \sim \text{Bin}(n-1,p)$ with $n >1, p \geq 1/2$ and $X,Y$ are independent. I'd like to show
$$(X\mid X \geq 1) \succeq_{sd} 1 + Y.$$
Here $(X \mid \cdot)$ is the ...

**0**

votes

**1**answer

94 views

### Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...

**12**

votes

**1**answer

536 views

### Probability of two vectors lying in the same orthant

Let $S^{d-1} = \{x \in \mathbb R^d: \|x\| = 1\}$ denote the unit sphere in $\mathbb R^d$. Let $v$, $w$ be drawn uniformly at random from $S^{d-1}$, conditioned on their inner product being equal to ...

**-5**

votes

**0**answers

31 views

### find law for a sum of process

I want know how to find the law of
$$ \zeta_n = \frac{1}{n} \sum_{i=1}^N (\lambda_i+1)\sum_{j=1}^n (u_{ji}^\epsilon)^2$$
when n and N tend to infinity. $u$ is a fractional Ornstein Uhlenbeck process. ...

**0**

votes

**0**answers

60 views

### Large deviations type results for sum of i.i.d. random functions

Assume that $f_1, f_2, f_3,\ldots$ are i.i.d. random functions $[0,1]\mapsto \mathbb{R}$ such that
(1) random variables $M_k=\sup_{x\in[0,1]}f_k(x)$ have exponential tails,
(2) $f$'s are a.s. ...

**0**

votes

**1**answer

63 views

### Proving an inequation based on binomial distributions

Problem statement
Let $c \in \mathbb{N}$, $n_1 \in \mathbb{N}_0$, and $n_2 \in \mathbb{N}_0$ be integers and $p$ a probability. Furthermore, let $b(m,j,p) = \binom{m}{j}p^j(1-p)^{m-j}$ denote the ...

**8**

votes

**0**answers

70 views

### Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of ...

**1**

vote

**1**answer

57 views

### Hitting time of a stochastically continuous process [on hold]

Suppose $X$ is 1-d stochastically continuous process with $X(0) = 0$, i.e.
$X_s \to X_t$ in probability as $s\to t$ for all $t\ge 0$. Let $\tau = \inf\{t>0: |X_t|>1\}$.
[Q.] Is $\tau>0$ ...

**11**

votes

**1**answer

245 views

### Pursuit-Evasion type game on graph (“Flyswatter game”)

An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...

**2**

votes

**2**answers

82 views

### A question about Skorokhod embedding problem

The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...

**4**

votes

**1**answer

144 views

### Blumenthal and Kolmogorov 0-1 law

Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...

**6**

votes

**1**answer

215 views

### How to calculate expected value of matrix norms of $A^TA$?

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with ...

**3**

votes

**1**answer

145 views

### Smallest eigenvalue gap of a non-symmetric random matrix

The question: Let $A$ be the matrix whose each element is an independently generated random variable which is uniform on $[0,1]$. One can see that the eigenvalues of $A$ will be distinct almost ...

**1**

vote

**1**answer

86 views

### moment sequence which does not define a random variable vs convergence in distribution

I am encountering the following problem concerning existence of a limiting random variable (in distribution): assume a sequence of positive random variables $\{X_n\}_{n\geq 0}$ from which we know ...

**2**

votes

**0**answers

55 views

### What is the gap between the two defined distances from node to node in a random graph?

Give a sparse random graph $G=(V,E)$, every edge $(u,v)\in E$ is associated with a weight $w(u,v)$. We assume each $w(u,v)$ is geometrically distributed with parameter $p_{u,v}$. The weight of a path ...

**0**

votes

**0**answers

27 views

### Is it possible to estimate the Interaction information of three variables without knowing their joint distributions?

I want to have a measure of the "synergy" between two players in a game. Each player has its own win ratio (won/played), which I'm modeling as two binomial distributed random variables X and Y. A ...

**1**

vote

**1**answer

72 views

### Exponentially Bounded Sequence of Moments defining Distribution?

I have an exponentially bounded sequence $m_n = \lambda^n + c_n$ (i.e. the $c_n$ are quadratic in $n$) and would like to know if this sequence of moments defines a distribution. I considered applying ...

**1**

vote

**1**answer

108 views

### Two types of random walkers on square lattice

Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index ...

**1**

vote

**0**answers

47 views

### Modify exponential family representation to a semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space.
We ...