6,957 reputation
1444
bio website case.edu/artsci/math/mwmeckes
location Cleveland, Ohio
age 38
visits member for 5 years, 7 months
seen 2 hours ago

Assoc. Prof. at CWRU. My main research interests are in probability, geometry, and functional analysis.


2h
comment Higher Moments, what are they good for?
Essentially the same question was asked and answered here: stats.stackexchange.com/questions/2893/…
2h
comment Define “Mathematics Colloquium”?
A former department chair of mine used to say that if you asked colloquium speakers to prepare talks appropriate for undergraduates, then there was a chance some of the faculty would understand.
May
6
awarded  Popular Question
May
5
awarded  Notable Question
Mar
26
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment Functions that Calculate their $L_p$ Norm
How about $a=0$, $b=1$, $f(x) = 1$?
Jul
2
comment 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
comment 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
comment 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.