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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Expected finite time Queue Length in Birth-Death process

Consider a Birth-death process $Q(t)$ with reflection at zero with geometric arrival and departure. With probability $\lambda$ there is an arrival at each time-slot. At the end of each time-slot there ...
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12 views

Write a random variable as a convex combination of other 2 in order to prove convexity of Bernstein polynomials

I'm trying to prove that if $f:[0,1]\to\mathbb{R} $ is continuous and convex, then the Bernstein polynomials are too. (And I want to use probability theory to prove it) The hint that I've got is ...
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25 views

Basic results for chi square processes

I could not find any introductory material with basic results regarding chi-square processes. Their definition from The Supremum of Chi-Square Processes is as a sum of $d$ squares of independent ...
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129 views

Forbidden coin flips

Suppose I have a (possibly infinite) bag of coins with various weights. I select a coin and flip it $n$ times. Averaging over the choice of coins from the bag, there is some probability of seeing ...
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30 views

How possibly does a $\alpha$-stable process jump at this stopping time?

Lemma 2.3.2 of [Applebum2009] states that, If $X$ is a Levy process and let $\Delta X(t) = X(t) - X(t-)$, then $\Delta X(t) = 0$ almost surely for a fixed $t>0$. There is also a warning that, ...
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161 views

Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
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1answer
58 views

Sampling from random unimodular matrices of a particular type?

Is there a nice way to parametrize unimodular matrices of form $$\begin{bmatrix} a1& a2& 0& 0\\ b1& b2& a1& a2\\ c1& c2& b1& b2\\ 0& 0& c1& c2 ...
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probability for having the exact same result in a given number of test with a given number of possible combinations [on hold]

The questions was: What is the probability for having exact the same page at least twice when printing my whole SSD (as bytes) on paper. We have 1117bytes per page leading to ~6,3e2834 possible ...
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1answer
46 views

Number of samples needed as input to Bernoulli factory

Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...
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27 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 ...
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1answer
104 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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52 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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44 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
148 views

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

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? [closed]

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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27 views

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

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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31 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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123 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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59 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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28 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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139 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
47 views

Pairwise distance distribution for point clouds (normal distribution) [closed]

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
79 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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70 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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96 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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79 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 ...
4
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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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85 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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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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56 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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51 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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82 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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199 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
126 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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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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442 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 ...
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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) = ...
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185 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 ...
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299 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 ...
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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
101 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 $X_i$ ...
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1answer
59 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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216 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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91 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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46 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 $H>\frac{1}{2}$, ...