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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14 views

Variance of a functional of transition probabilities using spectral gap of Markov chain

Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite state space $S$ whose Markov kernel is $K$ and unique stationary distribution is $\pi.$ Then, reversibility ...
2
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1answer
78 views

Are most random variables trivially sub-gaussian? [on hold]

I'm trying to understand sub-gaussian RVs to see if they could be relevant to my work. The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform ...
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40 views

concentration inequalities for quadratic forms of correlated random vectors

Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...
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39 views

Stationary distribution of two-dimensional Markov Process?

A two-dimensinal Markov process $\{\theta_{t},S_{t}\}_{t=1}^{\infty}$ where $\theta_{t} \in \Theta$ and $S_{t} \in S$.$\Theta$ is a continuous state space and $S$ is a discrete state space. Suppose I ...
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1answer
144 views

Minimum number of people such that 2 can be expected to sit next to each other [on hold]

We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here ...
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40 views

How to find PDF of ordered random variables? [on hold]

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
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24 views

probability of reaching a point in a 2d grid in a certain number of steps [on hold]

I have a random walk process in a 2d grid with N steps where N is small. How can I calculate the probability that any given cell was reached in N, N-1, N-2 ... 1 steps. That is, I would like to be ...
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28 views

Statistical test for the independence of components in a N-dimension Gaussian random variable

Assuming that there is a N-dimenson Gaussian randon variable $\mathbf{X}=\left[\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_N\right]^T$ and we have K observations ($K\ll N$) of $\mathbf{X}$(i.e., we have ...
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110 views

Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A_1,A_2,A_3\in\Bbb N$ with ...
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54 views

Probability of correlated quadratic residues

Given $N,a,c\in\Bbb N$, where $a\in(0,1)$ is fixed, $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A,B\in\Bbb N$ such that $N^{a/2}(\log N)\leq A,B\leq ...
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27 views

convergence of empirical distribution of random vectors

Given (a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal entries $A_{ij}\sim \mathcal N(0, 1/n)$; and (b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...
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137 views

Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...
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2answers
45 views

Pairwise distance distribution for point clouds (normal distribution) [on hold]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$). My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
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2answers
77 views

Expected summation of dropped intervals?

For each $n\in\mathbb{N}$, let $I_n$ be an interval of length $1/2^{n}$. We drop each $I_n$ onto the interval $[0,1]$ uniformly at random (so that there is "wraparound" if need be). What is the ...
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63 views

Showing that a sequence of random variables with increasing expected value converges to a Poisson random variable

Consider a sequence $\{X_n\}_{n \geq 1}$ of nonnegative, integer-valued random variables. For any random variable $Y$ and $k \geq 1$, let $(Y)_k = Y(Y-1)(Y-2)\dots(Y-k+1)$ be the $k^\mathrm{th}$ ...
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94 views

Randomly partitioning the unit interval with continuous functions

I want to construct a family of continuous functions $H$ in order to randomly partition the unit interval. That is, consider a partition $\lambda$ of the unit interval into $n$ subintervals: $\lambda ...
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77 views

Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question whether or not this is true: If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least double that of its ...
3
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54 views

Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. ...
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58 views

Time varying markovian birth death process

In a markovian birth death process, birth rate and death rate are $\lambda(t)$ and $\mu(t)$ respectively with Poisson arrival and departure. The population at $t$ is defined as $N(t)$. The problem ...
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2answers
81 views

Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound?

If $X \sim Normal(0,1)$, then we have the tail bound: $$ (*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$ Now for general variables $X$, a nice condition is that $X$ be ...
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22 views

PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities: $$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$ $$ H = \sum_{k=0}^{M} 2 k \left[ ...
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55 views

product of random elliptic elements is likely to be hyperbolic?

In S. Carlip's paper "Four-Dimensional Entropy from Three-Dimensional Gravity"(http://arxiv.org/abs/1503.02981) he states " the product of a large number of random elements of ...
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50 views

Fractional Sobolev space as a Cameron-Martin space

In their paper "On Fractional Brownian Processes" (link here to the working paper), Feyel and Pradelle (1997) write in the introduction: "we give very simple proofs of the existence of different ...
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31 views

Deducing probability density functions from model equations [closed]

I need to code stochastic models: $x_{n+1} = f(x_{n},\theta)$ where $x_n$ is the state of my system at time $n$ and $\theta$ is a set of parameters for this model, constant through time, and ...
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47 views

Jackson's theorem to optimize mean queue length of a traffic model

The following question I posted in mathematics stack exchange one month ago[http://math.stackexchange.com/questions/1307077/jacksons-theorem-to-optimize-mean-queue-length-of-a-traffic-model] but ...
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81 views

Measurable selections of a finite familiy of measures

EDIT. I'm adding a missing hypothesis and a really TL;DR version of the core problem. Warning: This short statement is the strongest form of what I want, hence not as plausible as the original form. ...
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44 views

Understanding the asserted variation in a WASEP particle model

Gartner's paper I hit a real issue on page 239 if anyone here understands what's going on? This page explained itself up until the statements (3.7) about the quadratic variation of $M$
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1answer
198 views

A question about brownian motions

I would like to ask a question about Brownian motion: Let $B$ be a standard brownian motion. How to show that $\mathbb P( \max\limits_{0 \leq s \leq t} B(s) \in (a,b) )$ decreases exponentially in t ...
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1answer
124 views

Gradient of Probability Distribution

Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ ...
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1answer
40 views

Reflection “monotonicity” of two point function percolation

Consider two dimensional anisotropic Bernoulli percolation where the parameter in the vertical direction, $p_v$, is strictly larger than the parameter in the horizontal direction, $p_h$. Suppose that ...
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11 views

Defining connectivity between K points on a periodic domain in terms of proximity

THE SITUATION: Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then ...
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1answer
438 views

Is the set of the convolutions of two-point measures dense in the set of all measures?

A measure supported in two points is a measure of the form $$ \mu=\alpha\delta_a+(1-\alpha)\delta_b, $$ where $a<b$ and $\alpha\in (0,1)$. The question is: Given a finite non-negative measure ...
6
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1answer
113 views

Is there an $\infty$ version of the Wasserstein distance between two distributions?

If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the $p$-th Wasserstein distance between the two of them is defined as $$W_p(\mu,\nu) = ...
3
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1answer
180 views

Information theory from negative probability

Szekely provides a convincing argument of negative probability here: http://www.wilmott.com/pdfs/100609_gjs.pdf What does a reformulation of classical information theory built from negative ...
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170 views

Paths in Pascal's triangle; or balanced $0-1$ initial segments

Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...
4
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1answer
258 views

Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
48
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1answer
3k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
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1answer
99 views

Distribution of bounded summation of i.i.d random variables

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...
2
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1answer
58 views

Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is: $$ p(x,y)=p(x)p(y). $$ Suppose instead that we have conditionals. ...
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1answer
213 views

Another name for coin-flipping polynomials

In his paper Functions arising by coin flipping (section 4), Johan Wästlund coined the term "coin-flipping polynomial" for polynomials that arise in connection with observing a finite number of coin ...
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90 views

Inequality about moments of a random variable and of its conditional expectation

This is a follow-up to a question I asked earlier: Moments of a random variable and of its conditional expectation My claim turned out to be false. Here is a new claim. Let $X$ be a bounded random ...
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1answer
104 views

Recursive sequence of binomial random variables

Fix $p>0$ and define a recursive sequence of random variables with $X_1 =1$ and $$X_{k+1} = X_k + \text{Bin}(X_k,p).$$ Thus, $\mathbf E [ X_k ] = (1+p)^k$. I would like a left tail bound. Perhaps, ...
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1answer
59 views

More precise statement about lower bounds on the cover time of general graphs

Uriel Feige has shown in 1995 in his paper "A Tight Lower Bound on the Cover Time for Random Walks on Graphs", the following result: For any graph $G$ on $n$ vertices, and any starting vertex $u$ ...
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44 views

Stochastic integration with respect to Fractional Brownian Motion

I would like to know what can be said about integral process $X_t = \int_0^t e^{-sr} dB_s^H,t\in[0,\infty)$, where $B^H_t$ is Fractional Brownian Motion with Hurst parameter $H>\frac{1}{2}$, ...
7
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1answer
138 views

Moments of a random variable and of its conditional expectation

Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...
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1answer
114 views

Correlation between two distance measures on bitstrings

I have an infinite collection of $0/1$ random strings of length $n$ (i.e., say 010001110101), where each digit is an independent Bernoulli RV, with parameter $p_i$, $i:1...n$. Define the "trait ...
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26 views

The effect of a single Markov transition on fidelity

Let $p$ and $q$ be two probability vectors of length $n$. The fidelity (or Bhattacharyya coefficient) of $p$ and $q$, is $$ F(p,q) \ := \ \sum_{i=1}^n \sqrt{p_i \cdot q_i}. $$ Let $A$ be a ...
5
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1answer
102 views

Distribution of infinity-norm over the unit sphere

I need to compute probabilities of the form $P( \Vert X \Vert_\infty < r ),$ where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere ...
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141 views

Distribution of trivial subset sums

Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
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32 views

Ergodicity property for continuous-time Harris positive Markov process

I have posted this question on there, but got no answer. The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328: Theorem 13.3.3. If ...