All Questions
57
questions
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Centered Subgaussian Variables have better Properties
I am trying to understand the following proof:
Main Confusion: In particular, I am having a very hard time understanding the chain of inequalities in the proof for (3)': I think the first equality is ...
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1
answer
17
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Understanding the proof for Properties of Subgaussian Variables
Here are the definitions, statements and the proof that I am stuck on:
I am stuck on the last part of the proof where the author claims that setting $C = e$ automatically guaranties that (1) holds ...
- 593
1
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0
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23
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Confusion about LLN for Empirical Processes/Measures
I have seen the following LLN result on empirical measures:
(Statement 1) Let $X, X_1, X_2, \cdots, X_n$ be i.i.d. random variables taking values in $[0, 1]$. Then
$$
\mathbb{E}\sup_{f \in \mathcal{F}...
- 593
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42
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Why is Bernstein's Inequality legal here?
This is a follow up question based on this post: Bounding rows of random matrices.
I can not see why we can use Bernstein's inequality as mentioned in the answer. In particular, it seems like the ...
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votes
1
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68
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Probabilistic Combinatorial problem - consecutive runs of $1$s in a string of $0$s and $1$s
Let $n$ values $0$ or $1$ be arranged around a circle, and for a given $k$ consider the number of runs of $k$ consecutive $1$s. Suppose the $n$ values are independent and each of them is equal to $1$ ...
user1289888
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25
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Need Help Understanding the Proof of Lower Bound on Expectation of Maximum Gaussians
I am trying to follow the proof given here: http://www.gautamkamath.com/writings/gaussian_max.pdf. In particular, I would like to understand the following crude bound mentioned in the paper:
My ...
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1
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76
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Debug a flawed solution to a problem on independence and orthogonality
From Exercise $24$ in Tao's note: https://terrytao.wordpress.com/2015/10/12/275a-notes-2-product-measures-and-independence/comment-page-2/#comment-682699
Let ${X}$ be a random variable taking values ...
- 1,011
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29
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Sub-Gaussian Norm Basic Property: Infimum is Minimum?
This is the definition of a sub-gassian norm of a random variable $X$:
$$
\| X \|_{\Psi_2} = \inf\{ t > 0: \mathbb{E}\exp(X^2/t^2) \leq 2 \}.
$$
It is claimed that we have:
$$
\mathbb{E}\exp(X^2/\| ...
- 593
0
votes
1
answer
74
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Question about Minkowski dimension
I'm learning Minkowski and Hausdorff dimensions to study Brownian motion right now, and I'm trying to understand the reasoning behind the Minkowski dimension of the set $(0,1,1/2, 1/3,\ldots)$ being $...
- 303
1
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1
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37
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The Countable Collection of Measures on a Given Measurable Space Is a Convex Cone
I am proving the following statement:
Given any countable collection $\{\mu_{n}\}_{n=1}^{\infty}$ of measures on a measurable space $(\Omega,\mathcal{F})$, the summation $\mu:=\sum_{n=1}^{\infty}c_{n}...
- 5,930
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0
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145
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Prove that the Lebesgue function is continuous
I am self-learning probability theory from the text Measure, Integral and Probability, by Capinski and Kopp.
Exercise problem 2.3 asks to investigate the Cantor-Lebesgue function, show that it is ...
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1
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0
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57
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How they are shifting cover to non-cover space?
I was reading this paper.
I am unable to understand the blocked step. Can anybody help me understanding that step?
Proof of Theorem 4. Define $\mathcal{W}_L \subseteq \mathcal{W}$ by
$$
\mathcal{W}_L \...
- 456
0
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0
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46
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An unusual proof of the image measure theorem
This is Theorem 4.10, p. 101, of Probability Theory: A Comprehensive Course, by Klenke. I have written an unusual proof which I think/hope is correct, and I'm wondering if anyone has any comments on ...
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381
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Proving Conditional Cauchy Schwartz inequality.
currently I try to solve the following exercise $8.2.6$ in Klenke Probability Theory for fun.
It states the following:
Let $X,Y \in L^2(P)$ be random variables on $(\Omega,F,P)$ then prove the ...
2
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0
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A property of measure-generating function
I'm reading about measure-generating function (page 28) in Analysis III by Amann/Escher
Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be increasing and continuous from the left. We say that $F$ is a ...
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