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.
9,327 questions
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Query in application of Russo's formula in a paper by H.Duminil-Copin and Tassion
I was reading H.Duminil-Copin's and Tassion's famous paper A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model and I had a query about the application of ...
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Proving that $\Phi \left(\frac{j}{\sqrt{n}}\right)\leq \mathbb{P}(S_n\leq j)$ for $j\geq-2$, where $S_n$ is a symmetric simple random walk
Let $S_n$ be a symmetric simple random walk starting at $0$. Numerically, I verified that for any $j\geq -2$ such that $n+j$ is even, we have
$$\Phi \left(\frac{j}{\sqrt{n}}\right)\leq \mathbb{P}(S_n\...
- 753
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Donsker-Varadhan duality in conditional sense?
A coherent risk measure named Entropic Value-at-Risk was introduced as follows: Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space, $X$ be a random variable and $\beta$ be a positive ...
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Is $\mathcal C(\mathbb R_+, \mathbb R)$ measurable in the cylindrical $\sigma$-algebra $B(\mathbb R)^{\otimes \mathbb R_+}$?
Let
$$
\mathcal A(\mathbb R_+, \mathbb R) = \{ f : \mathbb R_+ \to \mathbb R \text{ a map}\}.
$$
For each $t \ge 0$, denote by
$$
\operatorname{ev}_t : \mathcal A(\mathbb R_+, \mathbb R) \to \mathbb R,...
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Convergence of Kaplan-Meier estimator for pooled sample
I am new to survival analysis. Recently I have been thinking about the Kaplan-Meier estimator for pooled sample. Suppose we have two group of samples, group 1 has $n_1$ samples from the survival ...
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Nemytskii operator on $L^2$ space
Let $(\Omega,\mathcal{F},\mu)$ be a measure space and consider a function
$f \colon \mathbb{R} \times \Omega \to \mathbb{R}.$
For the problem I work on, a seemingly good hypothesis to place on $f$ is ...
1
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1
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Integrability of a shifted version of a regular conditional expectation
Preliminary note: I have previously asked the following question on Math StackExchange too, but since it generated little interaction there, I decided that it maybe could be more suitable for this ...
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Log concavity of a Gaussian function
Fix $t > 0$ and consider the map
$$
f(x) = \log \mathbb{P}\{|\sqrt{x} + Z| \leq t\},
$$
where $Z$ is a standard Normal random variable on the real line.
Is it true that $f$ is concave on the ...
- 542
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Inversion of conjugate to cumulant generating function?
$\newcommand{\eps}{\varepsilon}$Let $X$ be a mean-zero scalar-valued random variable with cumulant generating function $f(t) = \log \mathbb{E} e^{t X}$, where $t \in \mathbb{R}$. Let $f^\ast$ denote ...
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The density of the direction of a random interval in a convex body
Let $K\subset \mathbb{R}^n$ be a convex body. We uniformly choose two points $X,Y$ in $K$ and denote the direction of $X-Y$ as $u$, where $u\in \mathbb{S}^{n-1}$, and $f(u)$ is the density of $u$. We ...
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Imaginary Convergence Conjecture [closed]
Revised π–Imaginary Convergence Conjecture
Let π be the usual constant, and let the real number line be ℝ and the complex plane be ℂ.
Define a sequence of complex numbers by
zₙ = e^(i·π·rₙ) + πⁿ / n!
...
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Lifting of non-reversible Markov chains for convergence acceleration
Background and motivation
Let $(\Omega, \pi)$ be a finite state space with a stationary distribution $\pi$. Consider an ergodic Markov chain on $\Omega$ with transition matrix $P$ that is irreducible ...
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Question regarding this exercise on Cramér-Lundberg model with added Brownian motion
I've been trying to do this little exercise for the past couple of days but I really can't proceed at all. It regards calculating this: Given the classic Cramér-Lundberg processes with added Brownian ...
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Linear operations over finite (non-abelian) group
Let $G$ be a finite group (non-abelian), $S_1,S_2\subseteq G$, $S_1\cap S_2=\emptyset $ and $|S_1|=|S_2|.$
Let $L$ be a $k\times m$ matrix such that every row has exactly one 1 and one -1, other ...
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maximal displacement of a branching random walk
Consider a branching random walk given by the collection of i.i.d random variables $X(i_{0},\ldots,i_{t})$. Here $t \in \mathbb{N}$ and $i_{k} \in \{1,\ldots,n\}$ for any $k \in \mathbb{N}$. Each $X(...
