**0**

votes

**0**answers

3 views

### Non-normality of limit of random variables

I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that
...

**2**

votes

**0**answers

20 views

### Construct a sequence of i.i.d random variables with a given distribution function, diagonalization?

Assume we have a sequence of i.i.d. random variables $X_1, X_2, \dots,$ on a probability space $(\Omega, \mathcal{F}, P)$ with$$P(X_n = 1) = P(X_n = -1) = {1\over2}.$$Given a distribution function ...

**-5**

votes

**0**answers

43 views

### Calculate the probability that the sum of the square of two integers selected random are divisible by 2 [on hold]

P((X,Y)|X^2+Y^2 is divisible by 2)X,Y are positive integers greater than 4 possibly equals

**1**

vote

**1**answer

103 views

### limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit:
$\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$
When ...

**2**

votes

**0**answers

51 views

### Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence).
Is it true that the associated sigma-fields satisfy ...

**-5**

votes

**0**answers

28 views

### Elementary Expected Value Question [on hold]

Let $N$ be a positive integer. A soon to be bankrupt
casino lets you play the game $G(N)$. In the game $G(N)$, you roll a
typical, fair, six-sided die, with faces labeled 1 through 6, up to $N$
times ...

**3**

votes

**1**answer

56 views

### Weak convergence of random variables in $L^2$ and vague convergence

Dumb question: Let $X_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$.
Suppose also that $\mu_n$, the distributions of ...

**3**

votes

**0**answers

46 views

### What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?

By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...

**4**

votes

**2**answers

125 views

### Brownian motion, quadratic variation, existence of partitions?

Let $B_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t_{k, n} : k = 0, 1, \dots, k_n\}$ $$0 = t_{0, n} < t_{1, n} < \dots < t_{k_n, n} = ...

**-4**

votes

**0**answers

60 views

### $\pi$-system and $\lambda$- systems [on hold]

I have some trouble with a theoretic-like exercise about measure theory, and I would like to have some help. The problem is stated in the book Mathematical Statistics, Jun Shao, exercise 5 of section ...

**2**

votes

**0**answers

82 views

### McDiarmid-like inequality for subgassian random variables

Let $X_n$ be a set of $N$ subgaussian random variables, not necessarily independent, with $E\exp(\lambda X_n) \le \exp(\lambda^2/2)$. Let $X=(X_1,\ldots, X_N)$ and $f:\mathbb R^N \rightarrow \mathbb ...

**-1**

votes

**0**answers

37 views

### core logic explanation needed [on hold]

there are 20 balls in a jar. Balls are categories into 4 colors 3 white, 10 green, 4 blue and 3 purple. One ball is drawn at random. What is probability of drawing another minimum number to make it ...

**3**

votes

**1**answer

85 views

### Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$

In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
It is not hard to see that for every $k < \infty$, and every $\epsilon ...

**3**

votes

**2**answers

107 views

### Deduce average order of $\phi(n)/n$ from probability that two integers are coprime

I've seen proofs of the fact that the probability of two random integers being coprime is $\frac{6}{\pi^2}$ (all of them leading to a use of the Riemann Zeta function and the Basel problem). In ...

**6**

votes

**1**answer

195 views

### Can we recover a topological space from the collection of Borel probability measures living on it?

Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?

**8**

votes

**1**answer

111 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**2**

votes

**0**answers

64 views

### Which self-reference restrictions can be weakened in probabilstic logic?

This work suggests that there is some generalization of Truth in terms of probability, which can be definable within the logic itself.
Is where any other thorems on self-reference restrictions, which ...

**2**

votes

**1**answer

76 views

### Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...

**6**

votes

**2**answers

222 views

### Generalized density functions on the natural numbers

If $a_1,a_2,\dots$ are IID random bits (correction as per Anthony Quas: these "bits" are $+1$ and $-1$ with equal probability), then with probability 1, the set of natural numbers $n$ such that ...

**2**

votes

**1**answer

327 views

### Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?

A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...

**4**

votes

**1**answer

158 views

### How to construct i.i.d. standard normal random variables on $\Omega = [0, 1]$ with the Lebesgue measure

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Then we can find independent random variables $X_1, X_2, X_3, \dots$ defined on $(\Omega, ...

**4**

votes

**1**answer

92 views

### Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...

**3**

votes

**2**answers

89 views

### splitting exponential random variable into independent components

$X$ follows Exponential $(\lambda)$. Can we split $X$ into two independent r.v.'s, i.e.,
do there exist functions $g$ and $h$ such that $g(X)$ and $h(X)$ are independent for any fixed $\lambda$? ...

**-4**

votes

**0**answers

37 views

### Superposition of functions [closed]

there are some load curves that can be shown as smooth random functions, is there any theory about superposition of these curves that forecasts peak amount of result aggregated curve in compare with ...

**3**

votes

**1**answer

45 views

### Reference request: Urbanik's work on random integrals and Orlicz spaces

Several important papers on Lévy processes are referring to the following paper:
K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces,
Bulletin de l'Académie Polonaise des Sciences, ...

**5**

votes

**1**answer

127 views

### Estimating entropy conditional to an event

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant ...

**9**

votes

**1**answer

89 views

### Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...

**7**

votes

**1**answer

95 views

### A case of nested central limits

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. We are interested in $N$ large and prime. The Fourier transform ...

**5**

votes

**1**answer

151 views

### Order between two completely monotone functions?

I am wondering if the following assertion is true:
Let $f,g:\mathbb{R}_+\rightarrow [0,1]$ be completely monotone functions on $\mathbb{R}_+^*$, that is, $(-1)^n f^{(n)}(x)\geq 0$ and $(-1)^n ...

**3**

votes

**0**answers

92 views

### Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...

**6**

votes

**1**answer

151 views

### Statistical independence and the Fourier transform

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. With the Fourier transform we can define $N$ random walk variables
$$
...

**1**

vote

**0**answers

48 views

### Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De ...

**20**

votes

**1**answer

373 views

### Why does McMahon formula look like the inclusion-exclusion principle?

The McMahon formula for the number of tilings of an $a \times b \times c$ hexagon by lozenges:
$$ \Big[H(a)H(b)H(c)\Big] \Big[H(a+b)H(b+c)H(c+a)\Big]^{-1} \Big[H(a+b+c)\Big]$$
looks oddly like the ...

**4**

votes

**1**answer

150 views

### Integers in Boxes Problem

Given positive integers $k$, $m$, $n$, with $m,n >> k$, suppose we have
$n$ boxes each containing $k$ randomly (uniformly) selected positive integers $x$ satisfying $1 \leq x \leq m$ (duplicates ...

**2**

votes

**1**answer

78 views

### First passage percolation for general graphs

There have been many questions about the behavior of first-passage percolation on specific graphs. In particular, it seems like cliques, grids, random graphs, and ladders are well-studied. But I can't ...

**2**

votes

**0**answers

71 views

### Number of self avoiding paths which are not ``tie together''

Consider the lattice $\mathbb{Z}^d$. Let $A_{n}$ be the set of returning self-avoiding paths (from $0$ to $0$) having length $n$. For any path $\omega \in A_{n}$, let $f(\omega)$ be the number of ...

**0**

votes

**0**answers

28 views

### How to determine if random variables are distributed according to a multivariate normal distribution? [migrated]

Suppose $(x_1, x_2, x_3)\sim N(\mu, \Sigma)$ where $\mu\in\mathbb{R}^3$ and $\Sigma$ is a $3\times 3$ covariance matrix, are the variables $A = x_1 + x_2$ and $B = x_2 + x_3$ necessarily distributed ...

**7**

votes

**4**answers

547 views

### fixed points of permutation groups

As is well-known (see, for example, a nice exposition by our own Qiaochu: https://qchu.wordpress.com/2012/11/07/fixed-points-of-random-permutations/) that the distribution of the number of fixed ...

**5**

votes

**1**answer

106 views

### Nearest neighbor for planar Poisson is normally distributed

This was previously asked on MathSE, but was not answered.
Answering a question, I realized that the nearest point for a planar Poisson point process (with constant intensity $\lambda>0$) is ...

**3**

votes

**0**answers

36 views

### $X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]

What are the SDE's satisfied by the following processes?
$X_t = B_t^q$
$X_t = (\sin B_t)^q$
$X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...

**2**

votes

**0**answers

62 views

### Distribution of Brownian local time at first hitting times of $1$ and of $\pm1$? [closed]

Here, $(B_t)$ is a standard Brownian motion, and $(L_t)$ its local time at $0$. Consider $$T=\inf\{t : B_t = 1\},\qquad\tau =\inf\{t : |B_t| = 1\}.$$
What is the distribution of $L_T$?
What is the ...

**0**

votes

**0**answers

57 views

### Summing up costs over a Markov chain

I apologize in advance if this question is too simplistic to be appropriate for MathOverflow. I have inquired in multiple places but have found little to indicate that this is a previously studied ...

**5**

votes

**0**answers

42 views

### Full distribution of FPTs in random walks on graphs

There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ...

**2**

votes

**1**answer

86 views

### maximal inequalities for dependent random variables

I want to know literature about maximal inequalities for dependent random variables i.e. upper bound for $P(\max_{n\ge k\ge 1}\sum_{i=1}^{k}X_i > \delta)$ where $X_i$ are dependent random ...

**4**

votes

**2**answers

166 views

### Maximum difference between heads and tails in absolute value

I toss a fair coin $n$ times. Some notation:
$S_i=$ difference between #heads and #number of tails after the first $i$ tosses, $1\leq i\leq n$.
$M_n=\max(S_1,S_2,\dots,S_n)$,
...

**0**

votes

**0**answers

59 views

### Is this probability distribution studied in literature?

Let $\theta_1,\theta_2,\theta_3$ be 3 non-negative random variables such that $\theta_1+\theta_2+\theta_3=1$ with the joint probability distribution
\begin{align}
...

**2**

votes

**0**answers

31 views

### Is there any known construction of IIC as a limit from supercritical phase?

Consider a nearest-neighbor percolation model on $\Bbb{Z}^d$, where each bond is occupied with probability $p$ independent of each other. Let $\Bbb{P}_p$ denote the corresponding law on the ...

**3**

votes

**1**answer

79 views

### $\int_0^t f(s)\,dB_s$ normally distributed, mean and variance

Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$
How do I see that $Z_t$ is normally distributed?
What is the mean and variance?
I need ...

**2**

votes

**1**answer

255 views

### Computing the probability of reaching any leaf of an $n$-ary infinite probability tree

Suppose you have two players $X$ and $Y$ fighting, both of which have $n\in \mathbb{N}, n\geq1$ life.
Each player has a probability $p_i$ of doing $i$ damage for all $i\in[0, n]$. Note that $p_0$ is ...

**9**

votes

**3**answers

366 views

### Probability that planar Brownian motion doesn't “encircle” 0

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^2$ and $T = \text{inf}\{t : |B_t| = 1\}$. Let $E$ denote the event that $0$ is contained in the unbounded component of $\mathbb{R}^2 ...