**1**

vote

**1**answer

43 views

### Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...

**0**

votes

**0**answers

18 views

### Variance of sums of correlated variables when sampling without replacement?

Background
Let $X_t$ be an alternating renewal process, a stochastic process on the state space {0, 1} where 0 = 'Broken' and 1 = 'Working'. Let $U$ be the cumulative sojourn working during an ...

**-5**

votes

**0**answers

49 views

### Exactly 2 Girls - Conditional Probability [on hold]

This is very confusing to me. I am really new with this stuff.
A couple wants to have 3 or 4 children, including exactly 2 girls. Is it more likely that they will get their wish with 3 children or ...

**0**

votes

**0**answers

60 views

### Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...

**3**

votes

**0**answers

51 views

### Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$):
$$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$
where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...

**2**

votes

**0**answers

29 views

### Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...

**6**

votes

**0**answers

92 views

### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...

**3**

votes

**0**answers

43 views

### Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and ...

**-2**

votes

**0**answers

29 views

### Odds of 7 straight spins on a roulette wheel falling within the same group of 12 numbers [on hold]

What are the odds of 7 straight spins on a roulette wheel rendering a number within the same group of 12 numbers? (i.e. 7 numbers within 1-12, 7 within 13-24 or 7 within 25-36). This would be a ...

**0**

votes

**0**answers

23 views

### Does a singularly perturbed cadlag process has sample paths in a Polish space?

In the theory of stochastic processes it is often said in the broader literature that Polish state spaces are the only important ones appearing in practice. Are there also examples of stochastic ...

**0**

votes

**1**answer

95 views

### On the inverse problem of Dobrushin

Dobrushin, in this paper, looked into the following problem. Suppose We are given a Markov kernel (conditional distribution) $P_{Y|X}$. Information theorist usually call $W$ a channel. It is known ...

**3**

votes

**2**answers

125 views

### Graph game minimum vertex degree

Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ >> log(n)$. Players are BR and MA (BR moves first):
BR claims an unclaimed edge from $E$, adds ...

**-2**

votes

**0**answers

71 views

### A basic doubt on the quantity $\ln E[e^X]$ [on hold]

I heard that the quantity $\ln E[e^X]$ expresses variance of $X$ other than $E[X]$. But, I can't prove it formally ? any help will be appreciated.
i.e. I want to see how $\ln E[e^X] \geq E[X]$ (other ...

**-4**

votes

**0**answers

31 views

### Probability of coin tosses [on hold]

I have a probability problem.
Given 14 coins what is the likelyhood that on one given toss of all 14 at once 7 land heads and 7 land tails?

**-1**

votes

**0**answers

17 views

### Probability of rolling the same number n or more times in m rolls of a k-sided dice [migrated]

So the only approach I can find to solve this problem is making computer simulations, anyone can explain a mathematical way to solve it? or recommend a book that can explain this topic.
thanks.

**0**

votes

**0**answers

42 views

### Independent Quasi Monte Carlo Sequences

I am generating some copulas with MonteCarlo and QuasiMonteCarlo sequences. In particular, I would like to generate a Student's t copula with QMC numbers.
Here is my problem: for Student's t copula I ...

**2**

votes

**0**answers

55 views

### Measurability for disintegration of a kernel

Let $(x, A) \mapsto P(x, A)$ be a probability kernel whose "target" (wikipedia terminology) is a product space $Y \times Z$, and say both $Y$ and $Z$ are compact metric spaces. For every $x$ there is ...

**4**

votes

**0**answers

163 views

### Navigation in a graph

The problem
Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$.
Assumptions about the graph: You may ...

**0**

votes

**0**answers

87 views

### Expectation of product of cosines [closed]

I posted this question on math.stackexchange (under the same title), but it has not received any answers. It is possibly more appropriate here in any case.
I am reading a paper that starts with
$$
...

**2**

votes

**0**answers

92 views

### Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?

Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$.
Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in ...

**-3**

votes

**0**answers

50 views

### How to verify a vector has a steady gradual increase mathematically? [closed]

I rank the values inside my vector then look to verify the values have a gradual but steady increase. Ideally if I had 10 numbers the smallest would occur first and the second smallest would rank ...

**2**

votes

**3**answers

293 views

+50

### Probability that a sum of intependent random variables hits a point

Let $X_1,\ldots,X_n$ be independent random elements of a normed space $X$. Suppose that $\sup_{x\in X}\mathbb{P}(X_i=x)=p_i$. What is the best known upper bound for
$$\sup_{x\in X} ...

**0**

votes

**1**answer

62 views

### Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed?
What is the distribution of euclidean norm of complex Nagakami?
Cheers!

**0**

votes

**1**answer

63 views

### Help in finding distribution of the following function of random variable

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$
$$X_2 \sim \mathcal{CN}(0,\sigma)$$
If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...

**1**

vote

**0**answers

40 views

### Rate of convergence in narrow convergence

Does anyone help me in the following question?
I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...

**12**

votes

**3**answers

596 views

+50

### Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game

The game
Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...

**3**

votes

**1**answer

94 views

### Ising model: probability of a long path of minus under plus boundary conditions

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions.
Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...

**0**

votes

**0**answers

68 views

### Green's function of the Ornstein-Uhlenbeck operator

Consider $\mathbb R^d$ with the Gaussian measure $d\gamma(x) = e^{\frac{1}{4}|x|^2}\,dx$. The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a ...

**2**

votes

**0**answers

71 views

### Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...

**3**

votes

**1**answer

124 views

### Real points of zero-dimensional real algebraic varieties

There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?).
Here is a closely related ...

**3**

votes

**1**answer

161 views

### Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...

**3**

votes

**0**answers

57 views

### How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...

**0**

votes

**0**answers

28 views

### Isotropic correlation function for a vector valued random field

I'm having trouble with some of the implications of the following theorem.
Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...

**0**

votes

**0**answers

108 views

### A probability application question

Suppose there are two possible states $H$ and $L$, with prior probability $p$ and
$1-p$ respectively. There are infinite rounds with a discount factor $ d$. In
round 1, you could choose a value ...

**0**

votes

**1**answer

170 views

### Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ [closed]

Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and ...

**0**

votes

**0**answers

13 views

### Dirichlet distribution: Normalization of alpha values [migrated]

I'm a programmer and currently trying to apply the Latent Dirichlet Allocation algorithm by Blei et al. on a text mining problem. I am using a library called gensim for this, which takes, among ...

**2**

votes

**0**answers

63 views

### A 1-D random variable from a random distribution

I have a random variable $X$ that is drawn from the pdf
$$
f(x; \mu, \sigma, \sigma_{\mu}, \sigma_{\sigma}) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{|\hat{\sigma}|\sqrt{2\pi }} ...

**13**

votes

**2**answers

621 views

### Parity of $\lfloor 1/(x y) \rfloor$ not equally distributed

A curious puzzle for which I would appreciate an explanation.
For $x$ and $y$ both uniformly and independently distributed in $[0,1]$,
the value of $\lfloor 1/(x y) \rfloor$ has a bias toward odd ...

**9**

votes

**1**answer

256 views

### Guessing the larger integer: A game-theoretic twist

The starting point for this question is the old chestnut, already discussed on MO, about a game show on which the host has chosen two distinct integers and the contestant gets to reveal one of them at ...

**0**

votes

**2**answers

89 views

### Asymptotics for Hitting the sphere from the Outside

The problem is: consider A a solid ball centered at 0 and the exterior starting point $x\in A^{c}$, what is the behavior of $P_{x}(T_{B_{r}(0)}>t)$ for $d\geq 3$ as $t\to \infty$,where ...

**4**

votes

**1**answer

263 views

### Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...

**2**

votes

**0**answers

72 views

### (Reference) Asymptotics of hitting probability by Brownian motion

The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" ...

**1**

vote

**0**answers

19 views

### Perturbing moments of multivariable distributions

Let $P$ be a multivariate probability distribution on $\mathbb R^n$ which is moment-determinate and let $\{m_k : k \in \mathbb N_0^n\}$ be the sequence of moments $P$. Fix an order $p$ and consider ...

**0**

votes

**1**answer

102 views

### Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...

**3**

votes

**0**answers

133 views

### Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...

**2**

votes

**0**answers

70 views

### hitting time of the first quadrant for a 2-d random walk

Suppose $(S_n)_{n=1}^{\infty}$ is a simple random walk in $\mathbb{R}^2$. Let $\tau_{(a,b)}$ be the hitting of the first quadrant when $S_0 = (a,b)$. Is there a way to compute or estimate the ...

**0**

votes

**1**answer

89 views

### Sum of n independent F distribution random variables [closed]

I need a help:
What will be the distribution of sum of $n$ independent F distributed random variables with parameters 1 and 1 (i.e., $F(x;1,1)$?
Formally, say $x_1,\ldots,x_n$ are i.i.d. as F(1,1), ...

**0**

votes

**0**answers

31 views

### Stability of simple conditions on functions under convolution and/or mixture

We consider families of smooth probability densities defined on $\mathbb{R}^+$, $p=(x\in \mathbb{R}^+ \mapsto p_n(x))_{n\in\mathbb{N}}=(p_n)_{n\in\mathbb{N}}$ satisfying
(i) $\int_{\mathbb{R}^+} ...

**13**

votes

**2**answers

362 views

### What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$

Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...

**5**

votes

**1**answer

485 views

### Publishing an elementary proof of a less-general and less-useful version of a classic result?

Background
Let $X_t$ be a stochastic process on the state space {Working, Broken}. Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's uptime). It is well-known ...