**-4**

votes

**0**answers

42 views

### Probability problem - no idea where to start [on hold]

I have been working on this question for a few days and I am completely lost on how to solve it. Any suggestions, comments, hints are greatly appreciated.
Participants are competing in a ...

**2**

votes

**0**answers

112 views

### Show that $SL_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at ...

**0**

votes

**0**answers

35 views

### Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the ...

**1**

vote

**1**answer

37 views

### A generalization of negative binomial distribution

Assume we have a set of n balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...

**1**

vote

**1**answer

72 views

### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...

**0**

votes

**0**answers

16 views

### Conditioned sum of n Poissons versus unconditioned Poissons

Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$:
...

**1**

vote

**0**answers

29 views

### Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...

**1**

vote

**0**answers

52 views

### Zero-one law in binomial random graph model $G(n,p)$

Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...

**0**

votes

**0**answers

57 views

### How to calculate gambling odds with house edge? [on hold]

Here is the outcome of gambling odds I am trying to find a formula to:
gambling outcome
So my question is: how do we get the Roll High Profits(40.01818), and Roll Low Profits(0.62727) from BET SIZE ...

**5**

votes

**1**answer

247 views

### Distributional equation X+Y=2X

Let $X$ be a positive real-valued random variable. Let $Y$ be an independent copy of $X$ and assume that the equality $X+Y=2X$ holds in distribution. Does this imply that $X$ is constant?

**1**

vote

**0**answers

75 views

### A probability question related to combinatoric problem

I am trying to solve a combinatoric problem. The problem is the following:
There are A,B,C three types of people. There are totally N people arriving sequentially and make a choice between two boxes X ...

**0**

votes

**0**answers

32 views

### Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...

**3**

votes

**1**answer

127 views

### Range of random walk

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...

**0**

votes

**0**answers

14 views

### is any closed form relation that can state the error probability of code versus its variable and check node degree distributions?

In Low Density parity check code design, when bit (or frame) error probability of code is the objective of the design, we need a closed form relation between error probably (or even an approximate or ...

**1**

vote

**1**answer

65 views

### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...

**0**

votes

**0**answers

45 views

### Matrix concentration inequality

Let $X \in \mathbb{R}^{n \times d}$ be a fixed matrix and $W \in \mathbb{R}^{n \times d}$ be a random matrix with elements $w_{ij} = x_{ij} + \epsilon_{ij}$, where $\epsilon_{ij}$ are iid subgaussian ...

**0**

votes

**0**answers

17 views

### Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$.
Let $A \subset E$.
Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator.
I want to prove that $A$ is absorbing.
I know that it is ...

**0**

votes

**0**answers

36 views

### Limit probability of a complete bipartite random graph $G(n,n,p)$ is connected [closed]

I need to calculate the following probability limit for a complete bipartite random graph $G(n,n,p)$ in the Erdos-Renyi model:
\begin{equation}
\lim_{n\rightarrow\infty}\mathbb{P}[G(n,n,p) \text{ is ...

**0**

votes

**0**answers

34 views

### Weak solutions and strong solutions of SDE [closed]

What is the differences and connections between weak solutions and strong solutions of stochastic differential equations ?
Thank you in advanced!

**0**

votes

**0**answers

127 views

### Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...

**7**

votes

**2**answers

148 views

### Is there a rate of convergence for Donsker's theorem?

For the standard CLT, one can easily estimate a rate of convergence if you assume that the random variables have a little more than two moments.
Let $S_n$ be the centered-scaled sum of $n$ iid ...

**0**

votes

**0**answers

25 views

### Feller property for Ito diffusion with Lipschitz coefficients

Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded ...

**4**

votes

**3**answers

343 views

### Why does the overhand shuffle converge to the uniform distribution on $S_n$?

Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...

**2**

votes

**0**answers

93 views

### markov processes and ergodic theory

For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...

**2**

votes

**2**answers

109 views

### Probability of no $k$ 1's in arithmetic progression in binary sequence of length $n$

It is well known [it's on Wolfram Mathworld, for example] that the probability of no runs of $k$ consecutive $1$'s will occur in a $\{0,1\}$-valued sequence of length $n$ is exactly equal to ...

**0**

votes

**1**answer

114 views

### What is the relationship between $E(X\mid\mathcal{A})$ and $E(X\mid A)$?

This question seems obvious, but not sure how to prove it.
Let $\mathcal{A}$ be a $\sigma$-algebra, and $X$ be a random variable.
Suppose $E(X\mid A)\le1$ for any $A\in\mathcal{A}$, can we conclude ...

**1**

vote

**1**answer

131 views

### Difficulty with a formula for a probability related to card shuffling

I've been reading this article on the overhand shuffle. In it the author uses a simplied mathematical model of the shuffle:
Pemantle’s model for the overhand shuffle is
parameterized by a ...

**7**

votes

**1**answer

194 views

### Is this simple-looking moment inequality true?

Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$,
$$
\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p ...

**1**

vote

**0**answers

62 views

### formula for density of maximal Poisson disk sampling of radius 1?

Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ...

**4**

votes

**1**answer

91 views

### Probability of existence of a base in the span of sparse vectors in GF(2)

For $i=1,2,\dots,l$, let $\mathbf{v}_i =(v_{i1},v_{i2},\dots,v_{in}) \in \mathbb{F}_2^n$ be a sparse vector in GF(2) such that all $v_{ij}$'s are independent for all $1 \le i \le l, 1 \le j \le n$ and ...

**2**

votes

**0**answers

35 views

### Existence of probability distribution satisfying upper/lower bounds on events

Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...

**0**

votes

**0**answers

34 views

### Prokhorov convergence of Gaussian measures

Consider a Hilbert space $\mathcal{H}$ and a sequence of centered Gaussian measures $\mu_n$ on it. The covariance operators of $\mu_n$ are defined via their eigenpair(eigenbasis and eigenvalue)) as ...

**16**

votes

**2**answers

417 views

### What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order?

What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order?
Here is the algorithm for drawing a sample from ...

**5**

votes

**1**answer

117 views

### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial ...

**4**

votes

**1**answer

53 views

### Concentration of U-statistics for exchangable distributions (and the unbounded case)

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If ...

**1**

vote

**0**answers

60 views

### The role of absolute continuity in stochastic ordering defined over sets of probability distributions

This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ ...

**1**

vote

**1**answer

156 views

### What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?

Let $X$ be a random variable with $|X|\le1$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?
Existing results:
It has been known that ...

**3**

votes

**1**answer

118 views

### Solving algebraic recurrence relations on a cyclic graph

I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms:
$p_i = 0$.
$p_i = 1$
$p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all ...

**3**

votes

**2**answers

128 views

### Moving a result from the unconditional to the conditional

I'm generally wary when lifting a result stated unconditionally to a situation where I'm conditioning on a random variable. Consider the following classical result in weak convergence:
Theorem. Let ...

**5**

votes

**1**answer

103 views

### Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...

**2**

votes

**0**answers

160 views

### Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process
$$
dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0
$$
where $a\in (-\infty,+\infty), ...

**0**

votes

**0**answers

37 views

### Quadratic characteristic and constancy

Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form
\begin{align}
\frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) ...

**5**

votes

**1**answer

131 views

### Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where
\begin{equation}
Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.
\end{equation}
To ...

**2**

votes

**0**answers

98 views

### Question about continuity in the “complete Skorohod Topology”?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process"
https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf
In one of the exercises, exercise 8.9 ...

**1**

vote

**1**answer

71 views

### Reference for a local density theorem for binary vectors

I have the following theorem written on my whiteboard, but have misplaced the reference. I believe the probabilistic method may be involved in the proof. Any pointers appreciated.
Theorem Let ...

**0**

votes

**1**answer

81 views

### Predictable quadratic Variation <.> has same intervals of constancy as the process

From
Revuz and Yor - Continuous Martingales and Brownian Motion 1999
Chapter IV Proposition 1.13
it is proven, that for a continuous local martingale $M_t$ the intervals of ...

**1**

vote

**0**answers

75 views

### Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the ...

**2**

votes

**1**answer

66 views

### Median of a uniform multinomial variable

Let $k\in\mathbb N^+$ be a positive integer.
Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$.
For $i\in ...

**3**

votes

**0**answers

56 views

### “Local” functional central limit theorem for the empirical distribution function

This question is a repost from Mathematics Stack Exchange, where it did not receive any answer.
Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb ...

**1**

vote

**0**answers

73 views

### Convergence of an rcll process along a random subsequence

I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...