**5**

votes

**1**answer

136 views

### Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...

**-2**

votes

**0**answers

54 views

### Finding an example for [on hold]

Let $\varphi$ be a periodic function s.t. at zero and every integer points it is equal to 1. Moreover it's equal to one in at least one point between each integer. Can we have two distinct density ...

**3**

votes

**0**answers

68 views

### Local time of Brownian motion + Lipschitz continuous function

Let $\mathrm{ Lip} (M)$ denote the space of all functions on $[0,T]$ with Lipschitz constant and $L^\infty$ norm bounded by $M$. Let $(B_t)_t$ be a Brownian motion defined on the probability space ...

**0**

votes

**1**answer

98 views

### A conditional expectation question about consecutive inner products

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each $v_i \in \{-1,1\}$ independently and with equal probability. Each $w_j \in \{-1,0,1\}$ independently with equal ...

**1**

vote

**0**answers

75 views

### How to show that two linear combinations of Bernoulli random variables have jointly Gaussian distribution (and more)

Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let ...

**1**

vote

**0**answers

45 views

### limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know
$$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$
where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...

**3**

votes

**0**answers

62 views

### Moments of random special unitary matrices

This should be both well-known and probably easy, but I was wondering if the following is known (and, if so, how to easily calculate the thing or where to read about how to calculate it):
what is ...

**-3**

votes

**0**answers

89 views

### Camel up Board game probability problem [closed]

There is this game called "Camel Up" which is basically about betting on camels.
I wanted to calculate probabilities that every camel has to end up 1st after 1 turn AND probability that every camel ...

**-1**

votes

**0**answers

35 views

### Equality in fraction of density [on hold]

For two densities $f_1,f_2$ which take value in $[t_{min},t_{max}]$ following equality holds
$$\frac{f_1(x+j)}{f_1(x)}=\frac{f_2(x+j)}{f_2(x)}$$
for all $j\in\mathbb{Z}$ and all $x\in\mathbb{R}$, ...

**0**

votes

**0**answers

49 views

### Local limit theorem for an infinite dimensional integer lattice

Can someone refer me to a local limit theorem for the sum ${\bf S} = \sum_{i=1}^n{\bf X}_i$ of a sequence of independent and identically distributed $d$-dimensional random variables $\{{\bf ...

**2**

votes

**0**answers

60 views

### Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$.
We know that $C_n$ has a giant component whose size is of ...

**-1**

votes

**0**answers

16 views

### Probability of guessing the colors of a deck of cards correctly [migrated]

10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or ...

**-2**

votes

**1**answer

158 views

### using jensen's inequality

Suppose we have an expression
f(x, h(x,y)), for some function f and h, and x, y are random variables,
now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's ...

**-1**

votes

**0**answers

31 views

### Batch Markov Arrival Process - Computing Ps(t) [closed]

Suppose to have a queue $Q$ that represents a finite size buffer.
We have multiple arrivals to the queue with the same arrival rate $\alpha$.
Every group that comes to the queue can have a maximum ...

**2**

votes

**1**answer

107 views

### Onsager-Machlup function and most probable path of a diffusion process

Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX_{t} = f(X_{t})dt + dW_{t},
\end{equation}
where $f \in C_{b}^{2}(R)$ is a ...

**0**

votes

**0**answers

48 views

### Question about the representation of Skorokhod

I have a question about Skorokhod's representation theroem. Let $\Omega=R^m$ and define the canonical process $X=(X_1, ..., X_m)$, i.e. $X(\omega):=\omega$ for any $\omega=(\omega_1,..., \omega_m)\in ...

**-2**

votes

**2**answers

69 views

### Convergence of series made out of Markov Chain

$\{X_n\}$ be a ergodic Markov Chain taking values in $\Bbb Z$. Can I find some sufficient condition under which the $E[e^{\sum_{i} |X_i|}] < \infty$ (or say with some high probability).

**1**

vote

**1**answer

84 views

### Large deviations for maximizer of random walk with drift

Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift?
Let $X_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a ...

**1**

vote

**2**answers

134 views

### Proving a random bipartite graph contains a perfect matching

I have the following problem
consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...

**6**

votes

**3**answers

365 views

### Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon

The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%.
This is (part of)
Pólya's theorem.
I have been looking for an analogous (numerical) result for the probability
...

**1**

vote

**1**answer

111 views

### Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process?

I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ):
let X being a stochastic process with almost surely continuous sample path, and such that, there ...

**1**

vote

**0**answers

73 views

### A natural sum over multisets (expectation over multinomial)

I think this is a natural question but am not sure where to find resources.
Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one ...

**-1**

votes

**1**answer

104 views

### Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$
...

**2**

votes

**0**answers

34 views

### Local time for drifted Brownian motion and comparison results for reflected diffusion

Suppose $X(t) = x+ \mu t + \sigma W(t)$ where $x\ge 0$, $\mu, \sigma>0$ are real constants, and $W$ is a standard Brownian motion. The Skorohod decomposition of $X(t)$ can be written as $Z(t) = ...

**2**

votes

**0**answers

103 views

### density of penalizations of Gaussian probability measures

Let $\mu$ be a Borel probability measure on $\mathbb{R}^d$. By following one of the standard proofs of Bochner's Theorem (mollify, use Fourier inversion and Levy's Continuity Theorem), it is easy to ...

**0**

votes

**0**answers

48 views

### Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...

**5**

votes

**0**answers

138 views

### A generalization of Jensen's Inequality

Jensen's inequality is well known as
$$E\big[f(X)\big]\le f\big(E[X]\big)$$
where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also ...

**10**

votes

**1**answer

261 views

### Does Brownian motion immediately visit both sides of a Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
...

**-3**

votes

**0**answers

50 views

### Establishing CDF of sup of Brownian motion and Brownian Bridge

Question 1: Let $W_t$ be a Brownian motion. Then how could we prove that $$\Pr\left\{\sup_t|W_t|<b\right\}=1-\frac{4}{\pi}\sum_{j=1}^\infty \frac{(-1)^j}{2j+1} ...

**-1**

votes

**0**answers

21 views

### Non homogeneous poisson process

I'm trying to model a chemical reaction using a poisson process but with a little tweaking. I want a rate $\lambda$ that depends on $X_t$ which is the quantity of one of the chemical compounds. For ...

**0**

votes

**1**answer

77 views

### Probability and Markov processes

Suppose I have a Markov chain (satisfying all conditions of ergodicity) that has a stationary distribution that is easy to sample from. ( Assume that we know the stationary distribution upto a ...

**3**

votes

**0**answers

60 views

### Reasoning about dependent and independent quantities by “degrees of freedom”

In his classic textbook Foundations of the Theory of Probability Kolmogorov defines Independence a little bit differenent then it is usually done today. He denotes a probability space by $(E, \mathcal ...

**1**

vote

**0**answers

50 views

### A series with long-tailed terms

Let's consider the following series:
$$
\zeta = \sum_{k=1}^{\infty} a_k \xi_k,
$$
where the sum is understood as the limit in $L_2(\Omega)$, $a_k \in \mathbb{R}$,
$\sum_{k=1}^{\infty} a_k^2< ...

**6**

votes

**2**answers

99 views

### Geometric interpretation of the average of two independent Cauchy distributions

Let me state two facts:
(1) It is well known that if one takes a point uniformly distributed on the unit circle, and then takes it stereographic projection, the corresponding measure induced on the ...

**0**

votes

**0**answers

24 views

### Coordinates Poisson Cluster parent point

Is there any method to know the position of parent point in 'Poisson Cluster Process'?
For information I use data with poisson distribution. data consist of (longitude, latitude, date).
I want ...

**-2**

votes

**0**answers

27 views

### Expectation of a function with a Gamma distributed random variable

Consider a truncated exponential distribution $F(x\left| \lambda \right.) = \frac{{ - {e^{ - \lambda x}} + {e^{ - \lambda }}}}{{ - {e^{ - 2x}} + {e^{ - \lambda }}}}$ on the interval $[1,2]$. The ...

**2**

votes

**1**answer

95 views

### Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
...

**0**

votes

**1**answer

63 views

### Extend product sigma-algebra to cross-constant sets

We have two probability spaces $(\Omega_1,\mathcal{F_1},P_1)$ and $(\Omega_2,\mathcal{F_2},P_2)$. Is it possible to construct probability space $(\Omega=\Omega_1\times\Omega_2,\mathcal{F},P)$ such ...

**2**

votes

**0**answers

75 views

### Probability question involving simulations of picking balls from a bag

I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...

**0**

votes

**0**answers

73 views

### What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in ...

**10**

votes

**2**answers

448 views

### Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...

**0**

votes

**0**answers

46 views

### Central limit theorems for unequal probability sampling (weak but ill-defined dependence)

Suppose we are choosing samples of size $s$ from a finite population
$\{a_1, a_2, \dots , a_n\}$
where our sampling is with unequal probabilities. Construct
$$
S_n = \sum_{k=1}^{n} a_k
$$
Under what ...

**0**

votes

**0**answers

77 views

### Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...

**1**

vote

**0**answers

51 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 $t\leq T$
...

**-1**

votes

**0**answers

79 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

58 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 ...

**3**

votes

**0**answers

90 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 ...

**1**

vote

**0**answers

42 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 ...

**8**

votes

**3**answers

285 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 ...

**5**

votes

**1**answer

310 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 ...