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      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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      41 views

      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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      33 views

      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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      1answer
      28 views

      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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      31 views

      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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      19 views

      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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      63 views

      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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      1answer
      91 views

      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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      21 views

      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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      1answer
      72 views

      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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      27 views

      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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      1answer
      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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      1answer
      85 views

      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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      1answer
      94 views

      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 ...

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