Tagged Questions

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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1
vote
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
3answers
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
1answer
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
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0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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 ...