**-2**

votes

**0**answers

16 views

### weighted restricted integer compositions and extended binomial coefficients [on hold]

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

**3**

votes

**0**answers

33 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 $\...

**1**

vote

**0**answers

22 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**

votes

**0**answers

56 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 $\rho$ ...

**4**

votes

**0**answers

102 views

### Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model.
For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete) random walk.
For the next level $\ell=2$, we ...

**5**

votes

**1**answer

139 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

101 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

81 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

52 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

128 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 ...

**3**

votes

**0**answers

75 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

63 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

94 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

279 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

**0**answers

54 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")...

**0**

votes

**0**answers

68 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

**0**answers

42 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 [on hold]

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 \...

**0**

votes

**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

37 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**

votes

**0**answers

21 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 ...

**-1**

votes

**0**answers

65 views

### $\mathbb{P}(X \geq nx) \sim \frac{1}{nx}$ implies $\mathbb{P}(X\geq x) \sim \frac{1}{x}$ [closed]

Let $X$ be a real-valued random variable. Assume that for every fixed $x>0$, $\mathbb{P}(X \geq nx) \sim \frac{1}{nx}$ as $n \rightarrow \infty$. Then $\mathbb{P}(X \geq x) \sim \frac{1}{x}$ as $x \...

**0**

votes

**2**answers

119 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

**0**answers

64 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**

votes

**0**answers

35 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

106 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

65 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

45 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

258 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

186 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

226 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

133 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

80 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

169 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 ...

**4**

votes

**0**answers

93 views

### Concentration of infinite-dimensional Gaussian measure

I have the question about finding the subspace of concentration of a Gaussian Measure. More precisely:
$\textbf{Question:}$ Assume we have a separable Hilbert space $\ell_2$ with Borel $\sigma$-...

**4**

votes

**2**answers

237 views

### Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?

In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$:
$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...

**7**

votes

**0**answers

55 views

### Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ be i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$
...

**1**

vote

**1**answer

76 views

### Measurable isomorphism between two non-totally ergodic systems

Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...

**1**

vote

**0**answers

35 views

### Existence of a proper entropy scaling for a discrete measure

In this (quite elementary) paper, the scaled entropy of an unbounded measure $\mu$ on $\mathbb{N}$ is defined by
$$
h_c(\mu) := \lim_{\epsilon \to 0} \limsup_{n \to \infty} \dfrac{H^\epsilon(X_n)}{c(...

**5**

votes

**2**answers

1k views

### Random walk to stay in an interval forever

Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities.
To ...

**1**

vote

**1**answer

123 views

### Question on a random vector

This relate to that paper:
http://www.stat.purdue.edu/docs/research/tech-reports/1982/tr82-17.pdf
Let $U_1,...,Un$ be iid uniform on (0,1).
Set $L_n=\max_{i\leq n} U_i$.
Also $S(n)= \inf\{i\leq n|...

**2**

votes

**1**answer

77 views

### Meaningful formalization of a continuum of Bernoulli random variables [closed]

I was wondering if there is a meaningful formalization for a continuum of Bernoulli random variables. Informally speaking, consider the interval $[0,1]$, and let's say that for every $x \in [0,1]$, ...

**3**

votes

**1**answer

77 views

### Tail bound for product of normal distribution

Let $U, V$ be two standard normal random variables with covariance $cov(U,V) = \beta \in [0,1)$. Let $W = UV$ be the product of two RV's, and $W_1, W_2, \ldots, W_n$ be n i.i.d copies of $W$, what's ...

**2**

votes

**1**answer

73 views

### Symmetry of concentration bounds on mean

Question summary:
If I have a two-sided bound, can I immediately get a one-sided bound with tighter constants?
Question details:
Let $\mathbf X = X_1,...,X_n$ be $n$ i.i.d. real-valued random ...

**1**

vote

**0**answers

59 views

### Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...

**1**

vote

**0**answers

134 views

### Order statistics of iid uniform RV and Pólya's urn model. Question about a.s. convergence

Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let
$$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$
be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there ...

**1**

vote

**0**answers

41 views

### Power spectrum of the difference of two Poisson processes with equal rates

I am studying the asymptotic properties of a dynamic a model involving the difference of two balanced Poisson processes (i.e., $\lambda_1 = \lambda_2$). I recently discovered the Skellam Distribution ...

**1**

vote

**2**answers

125 views

### Concentration of matrix norms under random projection.

Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables.
Are there cases where one can been able to quantify $...

**6**

votes

**1**answer

109 views

### Basic Definition and Notations in RWRE

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...