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1d
comment Difference in the Four Color Theorem
ummmm.... they're different because there are planar graphs with maximum degree exceeding 4.
2d
comment CLT for sums of an infinite sequence of rv with an asymptotic distribution
I believe that as long as you have independence, and the distributions are not becoming more and more crazy, there should be a version of the CLT. Some examples of this: dynamical central limit theorems (in these you don't have independence; just exponentially small dependence of $x_i$ and $x_j$ in $|i-j|$).
Feb
8
comment Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations
Is there any reason you expect a set of matrices like this to exist?
Feb
7
comment Special random variables and monotone class theorem
The fact that $F(x)=\int f(x,y)\,dm(y)$ is a continuous function of $x$ if $f$ is a continuous function of $x$ and $y$ is more elementary than the corresponding measurable statements. The author seems to want to do the core of the proofs at the elementary level, and then deduce the more general measurable statements essentially by density of continuous in measurable. (Maybe he/she perceives a pedagogical advantage that you see the meat of the proof in a more familiar context?)
Feb
7
comment Markov Modulated Markov Chain
I believe you're talking about a Hidden Markov Chain.
Feb
6
comment Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1 $
So here is a concrete example. Take $f$ to be $2^{-n}$ on $S_n$ for $n>0$ and $f=4/3$ on $S_0$. With a little more imagination, you could build $f$ that is $2^{-n}$ on each $S_n$ with $n>1$ simultaneously satisfying your 2 conditions. My argument shows that there are lots of these functions - the construction is very flexible.
Feb
5
comment Measure of intersections in probability spaces
There's a missing parenthesis in the inequality. It should be $\mathbb E\Big( \Big[ \sum \big(1_{A_i}-\mu(A_i)\big)\Big]^2\Big)$. It's positive because square numbers are non-negative.
Feb
5
comment mod 5 partition identity proof
Isn't each term of the right side the generating function of the number of partitions into $n$ terms of size 1 modulo 5?
Feb
5
comment Measure of intersections in probability spaces
Is this a homework question?
Feb
4
comment Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1 $
Clearly the exact same argument will produce a function that is locally constant around no point if instead of defining $f$ to be constant on the cylinder sets, one defines $f$ to be a locally constant multiple of $1+\sum_{n=1}^\infty x_n 2^{-n}$.
Feb
4
answered Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1 $
Feb
4
comment Divergence of general random series and a special case
Certainly not! If the $X_n$ take a value $A_n$ with probability $2^{-n}$ and 0 otherwise, then the series is convergent by Borel-Cantelli, no matter what the dependence structure is. At the same time, by building large $A_n$'s, you can exceed any desired moment bounds.
Feb
4
comment Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?
I was about to say that the series "should" converge using crude probabilistic heuristics: If $p_n$ is a prime, the next integers should have something like a $1/\log p_n$ probability of being prime. So the expected value of $1/(p_{n+1}-p_n)$ given $p_n$ should be something like $\log\log p_n/\log p_n$. Now the sum is something like $\sum \frac{\log\log n}{n\log^2n}$. Of course now I should check how this checks out in the Erdös-Nathanson paper.
Feb
4
comment Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?
Ummm... people don't like it when you use the imperative voice "Prove X". People do like it if you give context for the problem - how did it arise? etc.
Feb
2
comment Total measure and Riesz theorem
Is this a homework question?
Feb
1
revised What are the generating partitions of the odometer?
include the author's comments
Feb
1
comment Transition probabilities for the symmetric random walk on the integers
oops - thanks Ori for the fix.
Feb
1
answered Transition probabilities for the symmetric random walk on the integers
Feb
1
comment Hypothesis testing for independent and non-identical distribution
I think this problem does not belong on this site, but rather stats.stackexchange.com. It has nothing to do with the continuum hypothesis. You should probably explain what you mean by SPRT and generally give more background.
Jan
30
comment What are the generating partitions of the odometer?
Code a sequence by a's and b's according to whether it's in A or not. By looking at 4 consecutive terms, you can determine $x_1$; by looking at 8 consecutive terms, you can determine $x_2$ etc. The associated process is Toeplitz: it has a's every 2 terms; then b's every 4 terms filling some of the gaps; then a's every 8 terms filling some of the gaps between these b's etc.