# Questions tagged [pr.probability]

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

5,175 questions

**0**

votes

**0**answers

32 views

### On the distribution of a random point of a poisson process

Let $T = \{t_i\}_{i=1}^\infty$ be the set of points in a Poisson point process on the positive half-axis with parameter $\lambda$, $I \in \mathbb{N}$ be a positive integral random variable with ...

**3**

votes

**1**answer

105 views

### Why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?

I'm reading 《Algebraic geometry and statistical learning theory》.My problem is why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?...

**3**

votes

**0**answers

72 views

### Exchangeable Bernoulli random variables with bounded summation implies negative correlation?

Let $\big\{X_1, X_2, ..., X_n \big\}$ be $n$ jointly exchangeable Bernoulli random variables, i.e., exchanging the order of these random variables does not change the joint distribution. If we know ...

**3**

votes

**2**answers

153 views

### Probability of one species reaching zero before the other in a Markov process on a 2d lattice

$\textbf{Background}$: Say we've got a two-variable system of stochastic chemical reactions, with quantities $\vec{x}(t) = (x_1(t),x_2(t)) \in \mathbb{N}^2$ evolving according to the following system, ...

**-1**

votes

**0**answers

51 views

### Upper bound on expectation of product

I want to upper-bound the following quantity:
$$\mathbb{E}_Y\left[f(Y)g(Y)\right] $$
The idea would be to get something of the shape: $\mathbb{E}_Y[f(Y)]\cdot h(Y)$
where $h(Y)= j(\mathbb{E}_Y[k(g(Y))]...

**0**

votes

**0**answers

108 views

### Which probability distribution has the most outliers?

Let $k$ be a positive real number. Which probability distribution over $\mathbb R$ maximizes $P(|x-E(x)|>k\cdot \operatorname{std}(x))$?

**0**

votes

**2**answers

111 views

### Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...

**1**

vote

**0**answers

34 views

### If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?

Let
$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$
$(N_t)_{t\ge0}$ be a $\mathbb ...

**9**

votes

**1**answer

278 views

### Maximal inequality for the average of i.i.d. random variables

Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like ...

**1**

vote

**0**answers

43 views

### Is there a transient graph whose spectral dimension two?

Let $G = (V(G), E(G))$ be an infinite connected simple graph.
Let $((S_n)_n, (P^x)_{x \in V(G)})$ be the simple random walk on $G$.
Let $p_n (x,y) = P^x (S_n = y)$.
A spectral dimension of $G$ is ...

**1**

vote

**0**answers

55 views

### Construction of Feller's pseudo-poisson process

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\...

**0**

votes

**0**answers

81 views

### Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.
...

**1**

vote

**0**answers

50 views

### If $P \ll Q$, are the regular conditional probabilities a.s. absolutely continuous?

Let $P$ and $Q$ be probabilities on $(\Omega, \mathcal{A})$, and let $\mathcal{F}$ be a sigma-subalgebra of $\mathcal{A}$. Assume $P \ll Q$. Assume that $P(\cdot \mid \mathcal{F})$ and $Q(\cdot \mid \...

**5**

votes

**4**answers

164 views

### Concentration of closed random walks

Consider a random walk $S_n=\sum_{i=1}^n X_i$ where $P(X_i=+1)=P(X_i=-1)=1/2$ with $n$ large. By Chernoff's bound we know that, for example, $\sum_{i=1}^{n/2} X_i=O(\sqrt{n})$ with high probability.
...

**-2**

votes

**1**answer

42 views

### Using common samples to numerically estimate pairwise equality of three random variables

Let $X,Y,Z$ be three discrete random variables which I can numerically sample. I need to numerically estimate the probability that $X=Y$ and the probability that $X=Z$. I would like to know whether ...

**0**

votes

**0**answers

30 views

### How can I solve a constrained optimization problem with a random number of decision variables?

Here is my problem. Let $A_t$ be a random variable with Poisson-Binomial distribution with set of success-probabilities $\{q_1^{(t)},\ldots,\,q_n^{(t)}\}$, with $t\in\{1,\,2,\,3,\ldots\}$, $n\in\...

**1**

vote

**0**answers

72 views

### asymptotic behaviour of principal eigenfunctions and Large Deviations

Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...

**2**

votes

**0**answers

98 views

### Random walk and comparing sums of Exponential random variables

Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...

**3**

votes

**1**answer

108 views

### Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...

**2**

votes

**0**answers

48 views

### General SLLN-like asymptotic mean concentration

Disclaimer: As I am not very knowledgeable of the field to which this question pertains, I will introduce a temporary terminology to convey the idea of my question at the risk of conflicting with ...

**2**

votes

**0**answers

43 views

### Sufficient condition for square root fluctuations of an ergodic sequence

Suppose I have a random sequence $\mathbf{X}=\{X_n\}_{n\in\mathbb{Z}}\subset \mathbb{R}^{\mathbb{Z}}$ that is ergodic with respect to translations. I am interested in a sufficient condition on $\...

**0**

votes

**1**answer

60 views

### Expected sum of chosen coordinates in a random subset of a Hamming hypercube

Let $S$ = $\{v_1, v_2, ..., v_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v_i$ = $\langle x^i_1x^i_2... x^i_d\rangle$ for all $i \in [1,n]...

**6**

votes

**2**answers

181 views

### Nonlinear boolean functions

Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...

**0**

votes

**2**answers

246 views

### Sum of independent random walks

Given two independent random walks $S$ and $S'$ with different distributions for the random variables $X_1$ and $X_1'$, I am interested in studying the conditions that make their sum either a ...

**1**

vote

**1**answer

67 views

### Generalization of inverse transform sampling

If X is a random variable over an arbitrary alphabet, is there a (deterministic) function f() such that X = f(U), where U is a uniform random variable over the unit-interval?

**0**

votes

**0**answers

72 views

### Expected value of eigenvalue of matrix

Let $A = (X_{ij})_{ij}$ a square matrix of size $n$ where the $X_{ij}$ are (discrete) real random non-negative entries. Denote by $\lambda_1(A) \geq \dots \geq \lambda_n(A)$ the (random) ordered ...

**5**

votes

**2**answers

209 views

### Probability of at least two of $n$ independent events occurring subject to some conditions

Given a set of independent Bernoulli random variables $\{x_1, \dots, x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\...

**6**

votes

**0**answers

100 views

### Collecting proofs of the birth of the giant component

I want to collect different proofs of Erdös-Rényi result on the double jump of the largest connected component on $G(n,p)$ (or in $G(n,M)$.
I know the original proof of Erdös-Rényi, the proof that ...

**1**

vote

**1**answer

54 views

### gaussian isoperimetric result for minimal measure under translation

Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$.
Let $S \subset \mathbb{R}^n$ be a ...

**4**

votes

**0**answers

189 views

### What happens in the martingale CLT if I norm by the conditional variance instead?

TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...

**1**

vote

**1**answer

51 views

### References for Hellinger distance/affinity involving mixture distributions

For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ ...

**2**

votes

**1**answer

144 views

### The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...

**3**

votes

**1**answer

160 views

### Concentration of a modified random walk

Let $\varepsilon$ be a number in $(0, 1)$, consider the following random walk on the real line $X^{(0)}, X^{(1)}, \dots$, where
$X^{(0)}=0$
If $X^{(t)} > 0$, then with probability $.5$, $X^{(t+1)...

**8**

votes

**3**answers

198 views

### Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...

**2**

votes

**0**answers

76 views

### Open problems in Monte Carlo Simulation [closed]

I want to know some open problems in Monte Carlo Simulation, which is being studied or in a stalemate.
Could you please give me some advice? Thanks a alot

**5**

votes

**1**answer

297 views

### Random pairs of commuting permutations

Let $\Omega_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma_1,\sigma_2,\tau_1,\tau_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma_j \tau_k = \tau_k \sigma_j$ for each ...

**2**

votes

**1**answer

129 views

### Gaussian sum VS Brownian motion

Given independent Gaussian $d$ dimensional vectors $G_i$,
Let $ \sigma^2_n=\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T$. $||\sigma_n^2||$ is norm of $\sigma_n^2$.
Is there a $d$-...

**3**

votes

**2**answers

228 views

### Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...

**6**

votes

**3**answers

430 views

### Randomly picking $k$ members of $\{1,\ldots,n\}$

Every day, I randomly pick a sample consisting of $k$ members of $\{1,\ldots,n\}$ where $k\leq n$. I stop as soon as every number of $\{1,\ldots,n\}$ has been picked at least once. Let $S$ be the ...

**0**

votes

**0**answers

32 views

### Expression for the Markov Chain CLT variance for an arbitrary initial distribution

Let
$(\Omega,\mathcal A,\operatorname P)$ and $(E,\mathcal E,\pi)$ be probability spaces
$(X_n)_{n\in\mathbb N}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)...

**2**

votes

**1**answer

46 views

### Lyapunov-type function in a non locally-compact space and boundedness of the average

Set-up and question.
Let $\mathcal{X}$ be a complete separable metric space which is not locally-compact. Let $V: \mathcal{X} \to [0; +\infty]$ be a function and $(X_t)_{t\geq 0}$ a Markov process in $...

**0**

votes

**0**answers

46 views

### quadratic variation on n-sphere

Is it true, and if so, is there an easy way to see that the quadratic variation of standard Brownian motion on n-sphere is $\leq$ t?
Note: I am a novice in stochastic analysis.

**2**

votes

**2**answers

89 views

### Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets

$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...

**3**

votes

**0**answers

107 views

### Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables:
$B^m_{1,1}$ $B^...

**0**

votes

**0**answers

15 views

### Compactness if Emery (semi-martingale) topology

The set $\mathscr{S}$, of semi-martingales is a topological vector space under the Emery topology on the space of semi-martingales. There has been some recent research on closures in this topology (...

**3**

votes

**0**answers

268 views

### Random walk on $\mathbb{R}$ with “sticky” origin

Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...

**2**

votes

**0**answers

74 views

### An Incorrect Construction of the Ito Integral

Let $B_t$ be a Brownian motion defined on the interval $[0,T]$, with underlying (filtered) probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$. Call a function $f:[0,T]\times\Omega\...

**2**

votes

**1**answer

83 views

### Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...

**0**

votes

**0**answers

59 views

### $L_1$ convergence for a product of indicator functions

Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions
$$
\lim_{N\...

**3**

votes

**0**answers

33 views

### product of right continuous filtrations is right-continuous?

Let $\mathcal{G} = \sigma \lbrace G_{1},..., G_{n} \rbrace$ where $G_{1},..., G_{n}$ are subsets of $\Omega_{1}$ and $(\mathcal{F}_{t})$ is a right-continuous filtration on $\Omega_{2}$. Is $(\mathcal{...