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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Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence: $$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$ and this ...
-3
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0answers
43 views

Find the joint density function?

Assume that $X_t$ is the OU process , i.e, $dX_t=\kappa(\theta-X_t)dt +\sigma dW_t$ where $0\leq t\leq T$ and $X_0=x_0>0$. Let $q(x)=\frac{\kappa}{\sigma}(\theta-x)x +\frac{\sigma}{2}$. I want ...
-2
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0answers
20 views

Branching process and process stochastic [on hold]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
10
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0answers
222 views

Probability that random nonnegative integer matrix is singular

Q. What is the probability that an $n \times n$ matrix, whose elements are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular? For example, for $n=3$ and $k=2$, the first ...
3
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1answer
93 views

Stationary distribution of last passage percolation

Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ...
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0answers
21 views

Spectral densities of stationary Feller processes with no diffusion, constant positive drift and negative jumps

For a (real valued, finite variance, centered) stationary process $X_t$ on $\mathbb R$, the auto-correlation function $k(\tau) = \mathbb E(X_{t+\tau}-X_t)^2$ and its inverse Fourier transform $\rho$, ...
13
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2answers
334 views

A probability distribution in n dimensional space which its projection on any line is a uniform distribution?

Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?
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1answer
119 views

Discretizing probability measures

Consider a probability distribution on $\mathbb{R}^k$, say $\mu$. Then there is a sequence of probability measures $\mu_n$ that converge weakly to $\mu$ so that each of them is discrete (takes ...
8
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7answers
556 views

Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
0
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1answer
318 views

Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
-1
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0answers
35 views

Probability/Bayes theorem/Bernoulli's experiments question [on hold]

John has rolled the dice 10 times and he said that every number form 1 to 6 has appeared at least once. Given this information find the probability of the event "number 6 has appeared at least 2 ...
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0answers
70 views

A Paradox by a Variant of Von Neumann's coin toss [on hold]

All biased coins are fair. If I have a biased coin whose probability of heads is $p$, and keeps tossing it, and only stops when the number of heads equals tails, then each sequence I get has a ...
1
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1answer
119 views

Arc Sine law for Random Walk conditioned to non-absorption or not?

Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$. Is the $E(S^{2}_{n}| \tau \geq n)$ ...
4
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0answers
113 views

inequality in a shape of inclusion exclusion formula

I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality: consider 9 numbers ...
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0answers
52 views

Random Cluster Model only for bond percolation?

Can someone please tell me which of the following statements I make are true of the current state of the art: The Random Cluster Model is a generalization of bond percolation (with possibly ...
6
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1answer
160 views

Length of nearest neighbor path in travel salesman problem

Given $n$ nodes uniformly distributed in $[0,1]^2$, consider the nearest neighbor algorithm to solve traveling salesman problem, i.e., each time I select the nearest neighbor not visited so far as the ...
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2answers
488 views

Given several beta distributions, what is the probability that one is the highest?

Given several random variables distributed according to different beta distributions, how can I calculate the probability that any one of those random variables is actually the highest? The ...
4
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1answer
171 views

continuity of the Boltzmann entropy in the Wasserstein metric

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ ...
6
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7answers
521 views
+50

Semicircle law universality elsewhere

Wigner's semicircle distribution is: $$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$ Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
0
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0answers
14 views

Bayes' Rule where the probabilities are taken as conditional [migrated]

I'm encountering some difficulty beginning statistics work with a basic Bayes' Rule problem. You can see the problem and answer on page 16 here, but I've explained it below. ...
8
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3answers
1k views

erfc lower bound

I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following? $$erfc(x) > \frac{ x ...
2
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1answer
123 views

Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like $$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$ Here $f(k,x)$ is actually a probability coming from a ...
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1answer
120 views

Is the following “section-wise” defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form Proposition: Assume that ...
2
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1answer
84 views

Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities

Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be ...
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1answer
66 views

Approximation of general measurable maps by simple functions [closed]

Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...
2
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1answer
56 views

Sub-$\sigma$-algebras and conditional expectation

Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space? In other words, is it true that conditional ...
4
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1answer
202 views

Properties of a finite random walk

Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise. Let $Y_N$ be the highest point $X$ have reached on the first ...
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0answers
19 views

Lower bound on the probability of guessing the mode in a small multinomial sample

Let $X=\left(X_{1},...,X_{k}\right)$ be a random variable that follows a multinomial distribution with $n$ trials and $k$ categories, with probabilities $p_{1},...,p_{k}$ such that $p_{1}-\delta\geq ...
-3
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0answers
17 views

Radial distribution and asset distribution [closed]

What are the classes of radial distribution that can describe asset value/returns distribution?
2
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1answer
545 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
3
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1answer
103 views

Expected number of leaf nodes in some theoretical graph models

If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of: (A) a random graph (e.g., Erdos-Renyi graph), (B) a small-world graph ...
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5answers
4k views

What is a cumulant really?

A cumulant is defined via the cumulant generating function $$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n},$$ where $$ g(t)\stackrel{\tiny def}{=} \log E(e^{tX}). $$ Cumulants ...
0
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1answer
57 views

Ergodic and mixing processes [closed]

I am working with an article, where it says: "that the discrete time stationary sequence $\{Y_j\}_{j\in Z}$ is mixing and hence ergodic." where $Y_t$ is defined as $Y_t = \int_{-\infty}^{t} ...
23
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0answers
2k views

When should we expect Tracy-Widom?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
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4answers
3k views

Constructing Bernoulli random variables with prescribed correlation

For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B_1, \ldots, B_n)$ with correlation $C$ ? Following the approach described in this MO thread, one can ...
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0answers
70 views

Why is this distribution exponential?

Take the interval $[0, 1]$. Now sample 10000 points in this interval randomly according to the uniform distribution. The fact is that the distribution of the distances between adjacent points on ...
5
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1answer
388 views

Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by: $X_{i}= h(i) + \varepsilon_i $, $h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$ where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...
2
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0answers
50 views

Stationary distribution for time-inhomogeneous Markov process

I have a two state, discrete time, time-inhomogeneous Markov process with transition matrix defined by $$T_i=\begin{pmatrix} 1-p_i\alpha & p_i\alpha \\ p_i\beta& 1-p_i\beta \end{pmatrix}$$ ...
2
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0answers
78 views

Tail bounds for suprema of random processes

Classical results concerning concentration of Gaussian random variables due to Cirelson, Ibragimov and Sudakov say that if $V_1,\cdots,V_n$ are jointly Gaussian with variance bounded by $1$, then ...
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0answers
47 views

Facebook Question (Data Science) [migrated]

Out of curiosity, here's a question from Glassdoor (Facebook Data Science Interview) You're about to get on a plane to Seattle. You want to know if you should bring an umbrella. You call 3 ...
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1answer
40 views

Common density of a random Vector with dependent entries ($Z=(aX,bX)$) [on hold]

Given a real random variable $X$ with density $g:R \to R$ and two real constants $a,b$. Let $Z:=(aX, bX)$, the random vector with entries as written (same random variable $X$ with different factors). ...
10
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0answers
230 views

Reference request: a combinatoric result [closed]

When I tried to construct a counterexample in my research, I encountered the following result, which should be true. Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = ...
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1answer
124 views

Proof of no bound for stochastic integral

I have Ito integral $X=\int_0^T f(t) dW(t)$ and I would like to proof that $P(X>K)>0$ for all $K$ provided $f(t) > \epsilon > 0$. My idea was $\int_0^T f(t) dW(t) \sim \int_0^T \epsilon ...
4
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1answer
254 views

Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived. Consider a ...
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0answers
31 views

How to Randomly Generate SAT Scores in R with max and min? [closed]

I'm trying to figure out how to randomly generate SAT scores in R by subsection. It follows the general form rnorm(n,mean,SD), but I also need to take into account that the minimum value has to be 200 ...
4
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1answer
75 views

Weakly correlated Bernoulli field

Let $\Lambda\subset\mathbb{Z}^{d}$ ($\Lambda$ is finite). Let $\left\{ \eta_{x}\right\} _{x\in\Lambda}$ be a field of dependent Bernoulli random variables. I assume that their correlation decays ...
2
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1answer
88 views

Probability of Hamming weight

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick ${n^t}$ random vectors $\{v_i\}_{i=1}^{{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$. Denote $v_j\cap v_j$ to be ...
2
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2answers
127 views

Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...
5
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1answer
246 views

Sums of random variables mod p

Let $\varepsilon_1, \ldots, \varepsilon_n$ be independent random variables taking values $0,1$ each with probability $1/2$. It is well known that $R_n=\varepsilon_1+ \cdots+ \varepsilon_n$ modulo a ...
2
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0answers
206 views

Hamming weight probability of projections

Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick $2^{n^t}$ random vectors $\{v_i\}_{i=1}^{2^{n^t}}$ such that each $v_i\in\{x\in\{0,1\}^{2^n}:|x|_{hamming}={2^{n-n^s}}\}$. If $v_i^\perp$ is ...