# 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.

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### Overview of interpretations of classical probability

The Stanford Encyclopedia of Philosophy has a nice overview of numerous different interpretations of probability (classical as opposed to quantum) with an extensive bibliography.
What books would ...

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### Mean field games approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.
I have a question about a step in theorem 3.8 on page 17. Let ...

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### Model for random graphs where clique number remains bounded

In the Erd?s-Rényi model for random graphs,the clique number is seen to go to infinity al the number of vertices grows. Is anyone aware of models for random graphs with bounded ...

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### Concentration or distribution of the scaled $l_p$ norm of a correlation matrix

Background:
Among Hermitan random matrices, correlation matrix has a lot of applications in statistics. People have studied the "empirical spectral distribution (ESD)" of a correlation matrix, the ...

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### Gaussian mean width of normal random cones

Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by
$$
w(T) := \mathbb E \sup_{x \in T} \...

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### Characterizing the relationship between element-wise Markov transitions and the full-conditionals of the stationary distribution

Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions.
One can ...

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### Concentration inequality for the law of iterated logarithm

The following question arose in one of my research projects. Before stating it, let me give a short background. We all know the law of iterated logarithm. It states that if $X_1,\ldots,X_n$ are i.i.d. ...

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### Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)

Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...

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134 views

### Theoretical framework for a divergent random series

Consider the following random variables
The $\{m_n\}_{n\geq 1}$ are iid and satisfy
$$\mathbb{P}(m_{n}\leq x)\leq C x$$
for $x>0$ and some $C>0$.
The $\{L_{n,m}\}_{m\geq n\geq 1}$ satisfy $L_{n,...

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### Conditional expectation of random vectors

$\newcommand{\E}{\mathsf{E}}$
$\newcommand{\P}{\mathsf{P}}$
The following additional question was asked in a comment by user Oleg:
Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B$ ...

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### Cross product of multi-variate Gaussians and their expectations

Let $a, b \in \mathbb{R}^3$ be two vectors, chosen independently from multi-variate Gaussian distributions ($a \sim N(\mu_a, \Sigma_a), b \sim N(\mu_b, \Sigma_b)$).
I'm trying to find a closed-form ...

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76 views

### Triangle inequality for Ito integral?

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...

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387 views

### A property about probability distribution

Suppose $g(x)$ is a pdf function and k is a positive real number. Let $F(\alpha)=\int_{-\infty}^{\infty}\frac{1}{\frac{g(x+\alpha)}{g(x)}+k}g(x)dx$, where $\alpha$ is positive.
I feel $F(\alpha)$ is ...

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### Sufficient conditions for inequality with integral of reliability functions

Let $Y$ and $W$ be two random variables with support $(y_1,y_2)$ and $(w_1,w_2)$ and distributions $F_Y$ and $F_W$, both twice continuously differentiable (densities $f_Y$ and $f_W$). Assume that both ...

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### Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...