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.
9,381 questions
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Stochastic order of generalized chi-square distributions (Extended)
I seek a generalization of the idea and answer found in
Stochastic order of generalized chi-square distributions.
Given that $χ^2_a=∑^n_{i=1}a_iZ^2_i$ where $Z_i∼N(0,1)$ are i.i.d. $\forall i$. The ...
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Random level sets for non-stationary random fields
Suppose I have a Gaussian, stationary and bandlimited stochastic process $s(t)$ (but hypotheses can be somewhat adapted) and a finite set of (fixed) sampling times $t_1, \dots, t_N$. Consider the ...
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Unboundedness of subsequences of cumulative sums of i.i.d. random variables
For natural $n$, let $S_n:=X_1+\cdots+X_n$, where $X_1,X_2,\dots$ are i.i.d. real-valued random variables. (We do not assume any conditions on the distribution of $X_1$ over $\Bbb R$.)
Let $(n_k)$ be ...
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Formula for cumulants with repetitions of $\pm1$ random variables
Let $X_1,\dots,X_n$ random variables taking values $-1,1$. Since $X_i^2=1$ we have
$$ k(X_1,X_1)\,=\,E[X_1^2]-E[X_1]^2\,=\,1-k(X_1)^2$$
and more in general one can express the cumulant where every ...
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A question about random walks on $ \mathbb{R}^2$
Let $\mu$ be a probability measure on $ \mathbb{R}$ with $\mu(\{ 0\})<1$. Otherwise, no further assumptions (e.g. on the existence of moments) are imposed on $\mu$. Let $X_n$ and $Y_n$ denote two ...
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A possible application of Stein lemma
Let $f:\mathbb R \to \mathbb R$ smooth, odd and positive on $\mathbb R_+$ .
I want to prove that
$$ \mathbb E[f'(W)] \geq0 $$
for a certain random variable $W$ which is a non linear function of ...
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More on a renewal process
The setup is part of that in the previous question, but the question is now different.
Let $X_1,X_2,\dots$ be independent random variables (r.v.'s) each uniformly distributed over the interval $[0,1]$....
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On a renewal process
Let $X_1,X_2,\dots$ be independent random variables (r.v.'s) each uniformly distributed over the interval $[0,1]$. For $k\in\Bbb N=\{1,2,\dots\}$, let
$$\tau_k:=\sqrt{X_k^2+X_{k+1}^2}.$$
Let $\tau_0$ ...
- 145k
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Extension of Strassen's Theorem, or existence of banded martingale transport
Consider two random variables $X\sim G$ and $Y\sim F$ on the real line. I would like to know if there exists a coupling of these with the following two properties:
$E[Y|X]=X$
$-a_L\leq Y-X\leq a_H$
...
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Limit distribution of particle position moving with random reflections
This is continuation of the question.
Let $G, \partial G$ be the interior and the boundary of simple polygon respectively. For $p \in \partial G$ let
$$
D_p=\left\{ d \in S^1: \exists\varepsilon > ...
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Limit uniform distribution of moving particle in simple polygon with random reflections
Let $G, \partial G$ be the interior and the boundary of simple polygon respectievly. For $p \in \partial G$ let
$$
D_p=\left\{ d \in S^1: \exists\varepsilon > 0 \text{ s.t. } p+\delta d \in G, 0 &...
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Does the additive channel model arise from average sum power constraint?
The Additive White Gaussian Model ($\mathsf{AWGN}$) model is the following: You send a message $x$ from a finite set of real alphabets $\chi$ and White Gaussian Noise (noise of Gaussian distribution $\...
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Financial Interpretation of the Lebesgue-Stieltjes Integral
I asked this question in the Quantitative Finance stack exchange (https://quant.stackexchange.com/questions/85294/financial-interpretation-of-the-lebesgue-stieltjes-stochastic-integral) and it was ...
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Is there a name for this type of probabilistic predictability of stopping times?
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space, and let $\tau \colon \Omega \to [0,\infty]$ be an $(\mathcal{F}_t)$-stopping time.
We will say ...
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Do we have a version of Hoeffding's inequality for these non-independent variables
I am writing a probabilistic argument (and I am not a probability theory expert), and the following would be useful to me. I tried asking AI but the answers did not seem helpful, so hopefully this is ...
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