bio | website | case.edu/artsci/math/mwmeckes |
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location | Cleveland, Ohio | |
age | 38 | |
visits | member for | 5 years, 6 months |
seen | Apr 29 at 13:05 | |
stats | profile views | 4,222 |
Assoc. Prof. at CWRU. My main research interests are in probability, geometry, and functional analysis.
Mar 26 |
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Is there an introduction to probability theory from a structuralist/categorical perspective?
@MartinBrandenburg: Moreover, I would need convincing that probability theory is about the category of probability spaces and measurable maps. That category surely plays a role, but most of what's of interest in the theory seems to take place outside of it. In particular, if you can show me how to view the law of large numbers or the central limit theorem as being about the category of probability spaces and measurable maps, then I'll reconsider. |
Mar 26 |
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Is there an introduction to probability theory from a structuralist/categorical perspective?
@MartinBrandenburg: I don't claim that it's distracting to define graphs as pairs of sets. I claim that if you do get distracted by the details of such a definition, then you're not doing graph theory. |
Nov 14 |
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Best possible concentration inequality in high dimensions
Ah, or by $\|X_i\|_2 \le 1$ did you mean that the Euclidean norm of $X_i$ is almost surely at most 1? I thought you were referring to the $L^2$ norm of a random variable. In that case, you can e.g. use a vector-valued version of Talagrand's inequality to get the same type of result as the one dimensional case. |
Nov 14 |
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Best possible concentration inequality in high dimensions
Yes, in 1 dimension this is Hoeffding's inequality if you have boundedness, but that was never mentioned. |
Nov 12 |
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Best possible concentration inequality in high dimensions
You need additional assumptions, even in the one-dimensional case, for Hoeffding's inequality to hold. |
Oct 22 |
awarded | Yearling |
Sep 30 |
awarded | Explainer |
Jul 17 |
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Clusters of uniformly distributed random points
@ChristianRemling: for the most part, MathSciNet only lists citations from papers indexed in MathSciNet beginning sometime in the 1990s, so it misses almost all citations older than that and many newer ones as well. (It looks like Naus's paper has lots of citations from papers in other fields.) |
Jul 3 |
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Nonexistence of stable random variables
I think I really should just read Feller straight through one of these days. |
Jul 2 |
awarded | Curious |
Jul 2 |
accepted | Nonexistence of stable random variables |
Jul 2 |
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Functions that Calculate their $L_p$ Norm
How about $a=0$, $b=1$, $f(x) = 1$? |
Jul 2 |
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Nonexistence of stable random variables
Here's a purely probabilistic proof that if nonzero $\alpha$-stable variables with finite second moment exist then $\alpha = 2$. Suppose that $X$ is $\alpha$-stable with finite second moment, and assume without loss of generality that $X$ is symmetric, so that $\mathbb{E} X = 0$. Then $$ \mathbb{E} X^2 = \mathbb{E} \left(n^{-1/\alpha} \sum_{i=1}^n X_i \right)^2 = n^{-2/\alpha} \left(\sum_{i=1}^n \mathbb{E} X_i^2 + \sum_{i\neq j} \mathbb{E} X_i X_j \right) = n^{1-2/\alpha} \mathbb{E} X^2, $$ and so $\alpha = 2$. |
Jul 2 |
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Nonexistence of stable random variables
@BillJohnson: That's a nice reduction. It becomes even nicer if you can take $p=2$, in which case you could even replace the (easy) facts about cotype with just the (utterly trivial) parallelogram identity. |
Jul 2 |
revised |
Nonexistence of stable random variables
added 398 characters in body |
Jul 2 |
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Nonexistence of stable random variables
Now one could ask the question, how do you prove that a stable distribution (as I defined it) must be $\alpha$-stable for some $\alpha$? But here I'm not interested in that; I really want to start with $\alpha$-stable distributions as I defined them above. |
Jul 2 |
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Nonexistence of stable random variables
@wolfies: Good point. Definition: $X$ has an $\alpha$-stable distribution if, whenever $X_1, \ldots, X_n$ are independent copies of $X$, there is a real number $d$ such that $X_1 + \cdots + X_n$ has the same distribution as $n^{1/\alpha} X + d$. |
Jul 2 |
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Nonexistence of stable random variables
I would say this is the "real" definition of a stable distribution. Versions given in terms of characteristic functions are only stated that way because the cf is the usual technical tool used to study them. |
Jul 2 |
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Nonexistence of stable random variables
@wolfies: Definition: $X$ has a stable distribution if, whenever $Y$ is an independent copy of $X$ and $a$ and $b$ are real numbers, there are real numbers $c$ and $d$ such that $aX + bY$ has the same distribution as $cX + d$. No characteristic functions in sight. |
Jul 2 |
asked | Nonexistence of stable random variables |