-4
votes
0answers
50 views

Best E-books and online-resources for Probability and its applications(especially games of chance) [on hold]

I am very much interested in studying games of chance and the probabilities related to our daily life instances but I need an online resource or some e-book to study them. I am a self-learner. Can ...
2
votes
2answers
220 views

Distribution of a random walk on a directed line

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line? $x_0 = n$ $x_t$ is a uniformly random integer between 1 and ...
1
vote
1answer
121 views

Finding loops and double edges ASAP in configuration model random graph

A common approach (at least in theory) to generating a random $n$ vertex graph uniformly subject to having a given (feasible) degree sequence $(d_i)_{i = 1}^n$ is to use the configuration model, i.e. ...
1
vote
1answer
135 views

Reference question: Brownian motion and surface area

I am doing research on the hitting probability of various sets (eg. 3D convex) and specifically how changes in perimeter/surface area change the hitting probability. By hitting probability I mean ...
1
vote
1answer
74 views

Minimum of Random Energy Model (REM) with logarithmically correlated potential

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...
4
votes
1answer
157 views

Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
4
votes
0answers
101 views

Nontransitive dice

In the wikipedia article https://en.wikipedia.org/wiki/Nontransitive_dice it is claimed that " The set of nontransitive dice were investigated by the Latvian computer scientist and mathematician ...
2
votes
0answers
62 views

Uniformization/measurable selection theorems

Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
1
vote
0answers
45 views

Jumps of jump diffusions

Let $W$ be a Brownian motion and $N$ a Poisson random measure defined on $\mathbb R_+ \times \mathbb R_0^n$ ($\mathbb R_0^n:=\mathbb R^n-\{0\}$) with compensator $\tilde N(dt,dz):= N(dt,dz) - dt ...
18
votes
2answers
693 views

Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...
2
votes
1answer
107 views

Concentration inequalities in $\ell_{\infty}$ for sums of iid random (“nice”) functions?

I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting): Let $D$ be a distribution on a set of "nice" functions ...
0
votes
0answers
36 views

Relation between Cardinality of Subset Weight-sums and the Weight's Number of Bits in Case of Random Integers

I would like to generate test-instances of "very general" finite, complete, symmetric graphs without self-loops and without parallel edges, which essentially boils down to: the edgeweights should ...
2
votes
1answer
161 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...
6
votes
1answer
243 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Here is what I mean exactly. ...
3
votes
3answers
208 views

Time-inhomogeneous Markov Chains

I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce ...
2
votes
2answers
169 views

How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where $A_i \sim \text{exp}(\lambda), S_i \sim ...
3
votes
0answers
106 views

Expectation of running maximum of diffusion processes

Let $X$ be a one-dimensional Ito diffusion $$X_t=x+ \int_0^t b(X_s)ds + \int_0^t \sigma(X_s)dW_s,$$ where $b,\sigma$ satisfy the usual Lipschitz continuity and linear growth conditions. Define the ...
0
votes
0answers
50 views

Linear Bounds on estimation error

Consider a markov chain on discrete state space $\mathbb{S} = \left\{1,2,..,S \right\}$, with transition probability matrix defined as $A = [a_{ij}]_{S \times S}$ where $a_{ij} = ...
1
vote
0answers
83 views

References on law of large numbers, CLT and iterated logarithm laws

Having access to those references, accumulating many results in one domain is always a bless, like Feller's book in probability, Dembo-Zeitouni's large deviation, Grimmett's percolation and recent ...
2
votes
1answer
122 views

Parameter estimation using bayesian update on moduli space?

Scientists take a set of data points, say in ${\mathbb R}^2$, and, assuming that this data should fit a polynomial of degree $d$ (or an exponential, etc.), they estimate parameters. I would think ...
4
votes
1answer
149 views

Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Queueing Model: Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) ...
6
votes
1answer
267 views

Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral $$ X_T:= \int_0^T W_t dt $$ It is often mentionend in literature that $X_T$ is a Gaussian with mean 0 and variance $T^3/6$. I am ...
1
vote
0answers
102 views

Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
1
vote
1answer
99 views

Convergence of a sequence of dependent binomial trials

Consider a sequence of the the following stochastic process. Let $b_0=1$, and let $n >1$. At each step $t$, let $b_t \sim Bin(n,\frac{b_{t-1}}{n})$. The process stops when either $b_t=0$ $b_t=n$. ...
3
votes
1answer
85 views

Random weighted selection without replacement

I am using the following procedure to select $m$ different numbers $\{i_1,\ldots,i_m\}$ from the set $\Omega = \{1,\ldots,N\}$, with $m,N\in\mathbb{N}$ such that $m< N$. Selection procedure ...
1
vote
0answers
158 views

An extrasensory perception strategy :-)

I asked this question at MSE some months ago but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a ...
6
votes
0answers
277 views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by ...
3
votes
1answer
109 views

The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$. This property is called the ...
4
votes
1answer
112 views

Connection between degree of growth and return probabilities of random walks on Lie groups

Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...
5
votes
1answer
172 views

When do iterated conditional expectations converge?

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X$ satisfying $\mathbf{E}[|X|]<\infty$. Define the iterated expectations of X as follows: $X_0 = X$, and, ...
1
vote
2answers
151 views

On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$ where $(t,x)\in {\mathbb R}_+\times ...
8
votes
3answers
412 views

References request: constructive quantum field theory

I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
5
votes
0answers
121 views

Semi-directed first-passage percolation on $\mathbb{Z}^2$ with deterministic vertical weights

Consider the following first-passage percolation problem on the plane grid $\mathbb{Z}^2$: all the horizontal edges are directed (pointing east) and carry an independent random weight, say standard ...
11
votes
0answers
479 views

Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...
0
votes
1answer
271 views

Stationary distribution in general Markov Chains

This is just a reference request for a result which is very general, useful and should be well-known, but I've failed to find a good reference to cite. The problem is to define the "most natural" ...
1
vote
0answers
67 views

Partially Observable Markov Decision Process - finding a hidden object with some positive probability

The following problem is example 5.1 from http://www.statslab.cam.ac.uk/~rrw1/oc/oc2013.pdf A hidden object moves between two locations according to a Markov China with probability transition matrix ...
4
votes
3answers
238 views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
6
votes
1answer
182 views

Finding cohesive (low exit probability) sets in a Markov process

The following is a fact about Markov chains that came up in a game theory paper. The purpose of this question is to ask if related notions or similar results are found elsewhere in probability, or are ...
1
vote
1answer
259 views

From Lebesgue Integral to Stieltjes Integral, and integration by parts

Let $X$ be a real random variable with c.d.f function $F$. Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite). What additional ...
0
votes
0answers
93 views

Master Equation to Fokker-Planck for a Jump-Diffusion

Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
1
vote
2answers
227 views

Asymptotics of the maximum of binomial random variables

Let $B_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In ...
1
vote
0answers
58 views

Small ball probabilities for functions of correlated normals

Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
3
votes
1answer
131 views

Concentration rates for the posterior distribution

Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution. What is known about the ...
7
votes
2answers
304 views

Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
2
votes
1answer
96 views

Linear combination of i.i.d. $Z_i$ distributed as $Z_1$

A classical property of the Gaussian distribution is that, if $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z_i \sim N(0,1)$) and $S = \sum_{i=1}^n a_i Z_i$ where ...
9
votes
2answers
268 views

“Fractional sampling” from a probability distribution

My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
4
votes
0answers
66 views

Estimates for simple random walks in groups of intermediate growth

I'm looking for references for the rate of escape and return probability for a group of intermediate growth. Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then ...
33
votes
1answer
2k views

Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown. I have already asked this question on ...
2
votes
0answers
167 views

Reference request: stationary measures as convex combinations of ergodic measures

Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures? I have found some references for the ...
7
votes
2answers
298 views

Measures whose projections are absolutely continuous

Since my question was not answered on MSE, I would like to ask it here. Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...