**0**

votes

**0**answers

26 views

### Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = ...

**2**

votes

**0**answers

136 views

### Is the integral always nonzero?

Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...

**0**

votes

**0**answers

21 views

### Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...

**1**

vote

**0**answers

45 views

### Poincare inequality for the measure of Brownian path

I am wondering if the Poincare inequality holds for the Brownian path space.
As the simplest example, let $\{w_t, t \in [0, 1]\}$ be a 1-d standard BM: has independent increments and continuous ...

**1**

vote

**0**answers

61 views

### Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$

Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.
Let $T_{M}, T_{N}$ be the smallest $n$ such ...

**-2**

votes

**0**answers

21 views

### calculate the mean and covariance functions of the stochastic process $X$ [on hold]

Let $X=\left(X_{t}\right)_{t\geq 0}$ a stochastic process given by:
$X_{t}:=A\cos\left(\varphi+\lambda t \right)$
where $\lambda>0$ is a constant, $A$ and $\varphi$ are independent random ...

**1**

vote

**0**answers

30 views

### Functions whose Laplace transforms have prescribed behavior at minus infinity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non-negative function with entire Laplace transform $\hat{f}$ (in particular $\lim_{t\to \infty}e^{st}f(t)=0$ for all $s$), and $p_0$ a positive integer. ...

**2**

votes

**0**answers

52 views

### convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on
a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$.
Suppose I know that
$$\int f d \mu_n \to \int f d\mu$$
...

**0**

votes

**0**answers

79 views

### Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...

**-1**

votes

**0**answers

32 views

### Random selection probability [on hold]

A test was given to a group of students. The grades and gender are summarized below:
...

**0**

votes

**0**answers

48 views

### interchanging limits for doubly indexed random sequences

I've encountered the following problem which seems to be quite standard but for which I can't find any proper references (asking on mathematics SE didn't bring up any answers so I'm reposting my ...

**6**

votes

**6**answers

429 views

### Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...

**3**

votes

**1**answer

58 views

### random odes adapted solution

Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode
$$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$
Where $b$ is a bounded continuous function (not ...

**12**

votes

**4**answers

670 views

### Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random ...

**3**

votes

**0**answers

160 views

### Unusual generalization of the law of large numbers

I have seen in physical literature
an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...

**-2**

votes

**0**answers

25 views

### Determining odds of a slot machine given a payout value of the icon [closed]

So most slot machines base the payout on the probability of the combination coming up. What I would like to do is flip that and set a payout and then have the probability based off of that if ...

**8**

votes

**2**answers

520 views

### random category theory

This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references ...

**0**

votes

**0**answers

61 views

### How to use Integrals to calculate the expected value of two-dimensional Gaussian distribution [closed]

Given that I have the following joint density function (two-dimensional Gaussian):
$f(u,v)= \frac{1}{1\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{-\frac{1}{2}Q(u,v)}$
where
...

**1**

vote

**1**answer

50 views

### Expected number of leaf nodes in some theoretical graph models

If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of:
(A) a random graph (e.g., Erdos-Renyi graph),
(B) a small-world graph ...

**1**

vote

**0**answers

51 views

### Probabilistic proof for expander existence [closed]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders).
I've been using the search ...

**0**

votes

**0**answers

66 views

### Alternate proof for Caratheodory extension theorem

This question is on the intuition behind the Caratheodory definition of measurable sets as given in Billingsley. He motivates by saying that we "should" call a set $A$ measurable if $$P^*(A) + ...

**6**

votes

**1**answer

96 views

### Does a Gaussian process shrink under a contraction map

Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X_t)_{t\in T}$ defined by $X_t = \langle G, t\rangle$, where $G$ is a standard ...

**3**

votes

**1**answer

115 views

### Regarding left-to-right minima

Let $\rho$ be a permutation on $[1,n]$ and $l_i$ be the number of left-to-right minima in $\rho_{i\ldots n}$, I know that for a random permutation $E[l_1] = H_n$ (the $n$-th Harmonic number) but is ...

**1**

vote

**1**answer

74 views

### Balls from bin with replacement, distinct elements, concentration inequality

Draw $n$ numbers, denoted by $a_1, a_2, \ldots, a_n$, from set $[n]$, that is, for each $i$, $a_i$ is a uniformly random number from $[n]$.
Let $A = \{a_1, a_2, \ldots, a_n\}$. Then
$$
...

**3**

votes

**1**answer

88 views

### Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed

The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n ...

**0**

votes

**1**answer

74 views

### Is the following “section-wise” defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form
Proposition: Assume that ...

**4**

votes

**2**answers

199 views

### First collision time of $n$ random walkers on a cycle

My question is somehow related to the one here First Collision Time for k Random Walkers on a Torus but, unfortunately, the answer does not cover my concern.
My problem is: consider $n$ walkers on ...

**11**

votes

**2**answers

251 views

### A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...

**-1**

votes

**0**answers

152 views

### Disprove this Piece of Jensen's Inquality “Black Magic” [on hold]

Jensen's inequality states that if a real valued function $f(x)$ is concave, like $f(x)=\ln |x|,$ then $E(f(X))\le f(E(X)).$ A classic application of this is $E(X) \le \ln |E(e^{X})|.$
Now consider ...

**1**

vote

**0**answers

28 views

### Equivalence of Graphical model selection algorithms

Suppose, a jointly Gaussian random vector is denoted by $X \in \mathbb{R}^{p}$ and $X$ has a distribution given by $\mathcal{N}(\mu,\Sigma)$. It is known that estimating the graphical model that ...

**0**

votes

**0**answers

13 views

### The mutual information rate spectrum [migrated]

Definition:
$\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...

**3**

votes

**1**answer

90 views

### General ballot theorem

I am looking for a version of the Ballot Theorem for general step distributions. Specifically, let $X_1,X_2,\ldots$ be i.i.d. real random variables with some distribution. Let $S_n = S_1 + \cdots + ...

**0**

votes

**0**answers

20 views

### product of two multivariate normal densities for the same vector, if one is only specified for a subset [migrated]

A random vector x with n elements has a multivariate-normal density f(x).
Another distribution is known for m linear combinations of elements of x. The linear combinations are given in the form ...

**4**

votes

**2**answers

256 views

### Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...

**7**

votes

**2**answers

213 views

### A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...

**6**

votes

**2**answers

192 views

### A variant of random walk

Standard random walk assumes a sequence of iid RVs $\{X_i\}_{i\geq 0}$ and studied the distribution of $S_n=\sum_{i=0}^n X_i$.
Here, I am wondering whether there is some work on
$T_n=\sum_{i=0}^n ...

**10**

votes

**0**answers

131 views

### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...

**1**

vote

**0**answers

47 views

### Bounding correlation between blocks of Gaussian stationary process

Let $X_n$ be a stationary Gaussian process with covariance function $\gamma(n)=\mathrm{Cov}[X(n),X(0)]$. Let $\mathbf{X}_p^q=(X_p,\ldots,X_q)$, $s_n^2=\mathrm{Var}(X_1+\ldots+X_n)$, and ...

**2**

votes

**2**answers

334 views

### Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that
the number of twin primes below $x$ should be roughly ...

**5**

votes

**0**answers

138 views

### Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...

**10**

votes

**4**answers

467 views

### Rate of convergence in the Law of Large Numbers

I'm working on a problem where I need information on the size of $E_n=|S_n-n\mu|$, where $S_n=X_1+\ldots+X_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX_1$. For this to make sense, the ...

**47**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**1**

vote

**1**answer

90 views

### Push-forward of sum of two maps

Let $X=R^n$ and $Y=R^m$ are two Euclidean spaces with $m<n$. Let $\varphi$ and $\phi$ are two (smooth) maps from $X$ to $Y$ and $\mu$ is a probability measure on $X$. Is there any relationship ...

**4**

votes

**1**answer

121 views

### Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...

**7**

votes

**1**answer

112 views

### Is there a Degenerate Dependency Local Lemma?

The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded.
Here I ask whether another possible generalization (for which I could not yet ...

**3**

votes

**0**answers

70 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
...

**8**

votes

**0**answers

193 views

### Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...

**2**

votes

**0**answers

48 views

### Bounds on number of distinct substrings

I have a table with $r$ rows of length $\ell$, with each cell containing a letter from an alphabet $A$ of length $a$. I'm trying to determine the expected number of distinct strings of length $k$ ...

**1**

vote

**0**answers

57 views

### Mixture with varying concentrations

Let $(\Omega ,\mathcal F, \mathbb P)$ be a probability space and suppose $$\mathbb P(X \in A) = H(A) = \prod _{i=1}^m H_i(A),\quad \forall A\in \mathcal F$$ be a distribution of a random vector $X = ...

**0**

votes

**0**answers

11 views

### A mix between the Horvitz-Thompson and ordinary estimator

I have two samples: unbiased $X$ with $N_1$ elements and biased $Y$ with $N_2$ elements from some distribution (let it be F = ChiDistribution(1) if needed, $N_1=N_2=50$).
Elements of $Y$ are picked ...