# Tagged Questions

**0**

votes

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15 views

### Mappings of random processes $\varphi(X(t))$

I am interested in problems of the following type.
Let $X(t)$ be a planar random process and $\varphi:\mathbb R^2\to\mathbb R^2$ be a mapping. Then what can we say about the image
$Y(t) = \...

**-1**

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**0**answers

33 views

### Deriving global probabilities from local dynamics

I am interested in growth dynamics and, in particular, how to derive difference/differential/stochastic equations from local behavior of a system. I tried posting first on math.stackexchange (https://...

**2**

votes

**0**answers

44 views

### Uniform mean-square-error estimates

Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of ...

**0**

votes

**0**answers

40 views

### Proving that a complex expression of integrals is increasing in a given parameter

Let $f$ and $F$ denote the respective pdf and cdf of a probability
distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real
$c$ such that $c\geq0$, and $\rho\geq0$.
We want to prove ...

**2**

votes

**1**answer

79 views

### Intuitional feeling of harmonic measure on one-third Cantor set

It is known that the harmonic measure on classical one-third Cantor set has Hausdorff dimension strictly less than $\frac{\log 2}{\log 3}$. Even harmonic measure has a close relation with brownian ...

**1**

vote

**1**answer

74 views

### Unique Stationary Distribution of A Markov Chain

I have a Markov Chain like $Y_i=\sum_n\pi_{n,i}(Y)Y_n$, i=1,2,3...N.
So the Markov chain has N states and the transition matrix depends on the vector $\textbf{Y}$. Moreover, $Y_i$ is continuous and ...

**3**

votes

**1**answer

108 views

### Randomly put $k$ balls in $2n$ circular boxes, pick $n$ consecutive boxes such that the number of balls is minimum!

You are given $2n$ boxes that are arranged circular (you can imagine all boxes are on the edge of a circular table). Then randomly, you put $k$ balls in the boxes such that each box is containing ...

**1**

vote

**1**answer

42 views

### Limit of stochastic subsequence of stationary ergodic sequence

Let $\{X_k\}_{k\in\mathbb{N}}$ be a stationary ergodic sequence on a probability space $(\Omega,\mathcal{F},P)$ with shift $T$. Also, let $\{v_k\}_{k\in\mathbb{N}}$ be a sequence of random variables ...

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**0**answers

63 views

### Advice on Family Index theorem [on hold]

I am reading Bismut's paper Family Index and Heat equation, but I have knowledge on the probability or stochastic.
Could anyone give some advice or introduce some ref. on probability to understand ...

**5**

votes

**1**answer

76 views

### Positive semidefinite ordering for covariance matrices

Suppose that X and Z are matrices with the same number of rows. Let
$$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...

**2**

votes

**2**answers

99 views

### Asymptotic Growth of Markov Chain

I asked the following question one week ago at math.stackexchange but didn't receive a response, so I want to give it here another try:
I'm interested in the following problem: We have got a time-...

**-1**

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**0**answers

33 views

### Probability - transformation [on hold]

I've just come across this derivation (it's only a fragment I'm interested in):
$$
\int p(x | \theta) \frac{\nabla_\theta p(x | \theta)}{p(x | \theta)} f(x) dx = \int p(x | \theta) \nabla_\theta \log ...

**-1**

votes

**0**answers

47 views

### Closure property of completely monotone functions [on hold]

A $C^{\infty}$ function $f(x_1, \dots, x_n)$ defined on $(0,\infty)^n$ is said to be completely monotone if
$$
(-1)^{k}\frac{\partial^k f}{\partial x_{i_1} \cdots \partial x_{i_k}} \geq 0
$$
...

**5**

votes

**1**answer

161 views

### A two-point inequality

Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...

**10**

votes

**2**answers

193 views

### Minimum separation among $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...

**1**

vote

**1**answer

84 views

### Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$...

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votes

**0**answers

23 views

### weighted restricted integer compositions and extended binomial coefficients [closed]

proof of
d_{S,f}(n,k) = \binom{k}{n}{(f(s)){s\in S}}

**6**

votes

**1**answer

241 views

### Stopping times for Brownian motion

Let $B_t, t\geq 0$ be standard Brownian motion.
Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$.
Define also a filtration $\...

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vote

**0**answers

39 views

### Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions
$p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...

**-1**

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**0**answers

63 views

### Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf
I will explain the general setup below.
Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...

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**0**answers

210 views

+50

### Hierarchical (Recursive) Random Walk (also known as Hierarchical Hidden Markov Model)

Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...

**6**

votes

**1**answer

149 views

### Random Cantor sets on the unit interval

Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$.
For ...

**0**

votes

**1**answer

108 views

### Necessary and sufficient conditions for Kolmogorov's Extension Theorem

Let $(X_n,\mathcal{X}_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y_n = \prod_{k=1}^n X_k$ and let $\mathcal{Y}_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod_{k=1}^\...

**2**

votes

**0**answers

86 views

### Threshold for prophet inequality

The prophet inequality is related to the following scenario:
Suppose there are $n$ independent positive random variables $X_1,\dots,X_n$. They might not be identically distributed. We reveal them ...

**2**

votes

**1**answer

55 views

### How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question.
Given a ...

**2**

votes

**1**answer

133 views

### Size of automorphism group of random regular graph

If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$?
--
I want to know what is the ...

**4**

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76 views

### Random polyominoes containing $2\times2$ squares

The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...

**2**

votes

**1**answer

65 views

### Version of Donsker-Invariance-Principle

Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...

**6**

votes

**2**answers

98 views

### Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables

Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...

**10**

votes

**3**answers

293 views

### How to sample a uniform random polyomino?

A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...

**1**

vote

**1**answer

81 views

### Reference on Probability theory on functional spaces (in special Hilbert spaces)

Currently, I am working on some sort of stochastic optimization problems defined over function spaces.
I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...

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votes

**0**answers

71 views

### Variation on stones in buckets

This is a spinoff, see Collecting stones in n buckets. Frankly speaking my only motivation is that I became curious: what happens if one redistributes the stones into the same buckets?
More ...

**0**

votes

**1**answer

48 views

### Conditions for supremum and conditional Expectation to commute

I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\...

**0**

votes

**0**answers

25 views

### Application of Lemma in Iterated Expectation [closed]

I was reading the following paper: InfoGAN.
I cannot figure out, how on page 4, Lemma 5.1 was applied in the following lines:
$$\mathbb{E}_{c \sim P(c), x \sim G(z,c)}[\log Q(c|x)] = \mathbb{E}_{x \...

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**0**answers

33 views

### A multifractal model of asset returns - Mandelbrot, scaling result

I am looking at the paper in the title and I am trying to derive their result in equation $2$. Here is what I obtain. Start with:
$$X(ct) \stackrel{d}{=} M(c)X(t)$$
where $M(.)$ and $X(.)$ are ...

**0**

votes

**0**answers

41 views

### Last Inference in proof of conditional limit theorem

I read about the Conditional Limit Theorem from the book "Elements of Information Theory" by Thomas M. Cover and Joy A. Thomas, second edition, page 371. I can't understand the last inference in the ...

**0**

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23 views

### Article Using Kullback Leibler Divergence to Measure Divergence of Observation from Distribution

I am currently attempting to compare an observed distribution to a theoretical distribution, and my current approach is to normalize the two and find the Kullback Leibler Divergence. I am beginning to ...

**0**

votes

**2**answers

120 views

### Coding SLEs (Schramm–Loewner Evolution) eg. SLE(6)

Any references/links on codes for SLEs written in C++ or Matlab that I can run in Windows (visual studio)?
The only code I found was:http://math.arizona.edu/~tgk/research.html but the link was empty. ...

**0**

votes

**1**answer

91 views

### Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...

**0**

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**0**answers

37 views

### Fisher metric for shift-invariant probabilities

I'm just discovering what seems to be the tremendous heuristic value of the (century-old, more or less) canonical Riemannian metric (Fisher metric) on the $n$-dimensional simplex $\Sigma_n:=\{(p_i)_{i=...

**1**

vote

**0**answers

109 views

### Probability that two integers selected from a fixed interval are relatively prime [closed]

I found the answer to a very similar question already asked here on mathoverflow: what is the probability that two natural numbers are relatively prime? The answer given in the link below was $\frac{6}...

**1**

vote

**0**answers

37 views

### Probability for a SRW to be at some place in an even number of steps

I am looking for some references for the following problem.
Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...

**2**

votes

**0**answers

66 views

### Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by,
$d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$
$X_0 = x$.
Suppose the functions $\mu$ and $\sigma$ are as follows -
$f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...

**1**

vote

**0**answers

46 views

### Expected amount of linearly dependent random vectors? [closed]

Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$?
EDIT: As said in the comments, I'm looking for the ...

**2**

votes

**1**answer

260 views

### Does random walk have more concentration surrounding the origin?

Consider a simple random walk $S_n$ on one dimension, starting at $0$. In this case, $S_n$ fluctuates between $-\infty$ and $\infty$, but intuition says that it might stay more often in an interval ...

**2**

votes

**0**answers

192 views

### Average minimum number of random k-sparse vectors in $\mathbb{F}_2^n$ to span a specific base vector?

A while back I posted a question in MO about the average minimum number of independent random k-sparse (having at most $k$ non-zero elements) vectors belonging to $\mathbb{F}_2^n$ to span the whole ...

**1**

vote

**0**answers

227 views

### Prove this function is increasing

I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...

**4**

votes

**1**answer

134 views

### Uniform sampling from general simplex with a twist

This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...

**3**

votes

**1**answer

83 views

### Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...

**5**

votes

**2**answers

171 views

### Frequency of visiting states in Markov chains

Given a finite, ergodic Markov $\{X_i\}$, and two natural numbers $a>b$. Let
$$p=P\left[\forall n, \sum_{k=n}^{n+a-1} \mathbf{1}_m(X_k)\leq b\right]$$
where $\mathbf{1}_m(X_k) =1$ if $X_k=m$ and 0 ...