cardinal
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Registered User
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May 11 |
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Chernoff bound in the not-quite-sub-exponential case Related: mathoverflow.net/questions/118562 |
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May 11 |
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Show that this ratio of factorials is always an integer This is just an observation: We can write the quantity as ${2m \choose m}{2n \choose n}/{m+n \choose m}$. The numerator is the number of random walk paths of length $2(m+n)$ such that the random walk is zero at time $2m$ and $2(m+n)$. The denominator hints at quotienting out the locations of pairs of steps instead of considering the first $2m$ and the subsequent $2n$. But, I haven't made that work out and suspect it won't. |
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May 10 |
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Inadmissibility of Simpson’s rule From Persi's website: P. Diaconis (1988), Bayesian Numerical Analysis, In Statistical Decision Theory and Related Topics, vol. 4, pp. 163-176 (www-stat.stanford.edu/~cgates/PERSI/papers/…). There is also a prior technical report from 1986 with the same title. (statistics.stanford.edu/~ckirby/techreports/NSF/…). You may also be interested in A. O'Hagan (1992), Some Bayesian Numerical Analysis, Bayesian Statistics, vol. 4, pp. 345-363 (stat.duke.edu/~fei/samsi/Readings/der0.pdf) and a related article on Bayes-Hermite quadrature. |
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May 10 |
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Concentration of sum of independent random variables Dear Mahdi, I appreciate the edits and I am trying to be helpful, so let me encourage you to reread my first comment and take it to heart. It now appears that all three of the issues mentioned in that comment are in play here. Regarding your most recent edit and comment: (a) Your edit is incorrect and does not match Prop. 5.16 or the associated Cor. 5.17, (b) my standard normal example is entirely relevant and intended, as you will see if you examine it more carefully and (c) though somewhat immaterial thusfar, a normal random variable is most certainly subexponential (as is its square)! |
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May 10 |
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Concentration of sum of independent random variables Yes, I am familiar with R. Vershynin's work; the way you gave the first bound was a dead giveaway that this was the paper you were looking at. :-) Consider the case $X_i \in \{-1,+1\}$ with probability $1/2$ each. Certainly this satisfies your conditions, but $n = S_2 \leq S^2$ is false, in general. (Take $n=2$, for instance.) As a second example, take $X_i$ to be standard normal and consider $n = 1$ to see that $\mathbb P(S_2 > t) \leq \mathbb P(S > \sqrt{t})$ is false even when $S_2 \leq S^2$. |
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May 10 |
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Concentration of sum of independent random variables Could you please take the time to revisit your post carefully and make some edits? Several statements are incorrect. I am unsure at this point if this is due to (a) some inadvertently unstated assumptions (e.g., that the $X_i \geq 0$), (b) multiple typos or (c) something more conceptual being missed here. I don't want to make any assumptions, so I'll leave it to you to make the needed edits. Cheers. |
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May 5 |
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Modern Mathematical Achievements Accessible to Undergraduates PRIMES is in P? |
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Apr 30 |
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Expected edit distance Undoubtedly I am suffering from lack of sufficient coffee intake, but I don't follow the "rigorous bound". Isn't $c_n := \mathbb E(E_n) / n \leq 1/2$ trivially by either (a) comparing to the Hamming distance or (b) using the subadditive property that Kevin mentions in his comment above? |
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Apr 29 |
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The fraction of the sphere a fixed distance from a subspace Are you looking for a uniform bound in $k$ and $d$? Otherwise, can you clarify what do you mean by a "best possible bound" (that wouldn't yield a tautological answer)? |
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Apr 27 |
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History of the high-dimensional volume paradox Regarding the first reference in that article: I can imagine almost no better name for the author of a lecture entitled An elementary introduction to modern convex geometry than K. Ball. |
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Apr 4 |
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Rapid evaluation of multivariate normal integral Some questions and remarks: Note that there is a high degree of symmetry here; in particular, the implicit joint probability is invariant wrt to $n+1$ iid standard normal random variables. Are there any restrictions on $a_i$ and $b_i$, or can they vary arbitrarily? Over what range of error bounds are you interested in? Are you looking for hard bounds or are you potentially content with "softer" statistical ones, e.g., via some Monte Carlo approach? |
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Mar 16 |
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eigenvalues of the sum of a stochastic matrix and a diagonal matrix Sufficient condition: $D = m I$ and $k > -m$. ;-) |
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Mar 14 |
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Concentration of sum of pairwise squared Euclidean distances of random vectors Notice that this is a (multivariate) $U$-statistic. I would start by searching with those terms. Have you already looked at the case $d = 1$? |
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Feb 26 |
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Is the Binomial Expectation of Convex Function Convex in p? (Furthermore, we can couple $X$, $X_1$ and $X_2$ by using the same sequence of iid $\mathcal U(0,1)$ random variables to generate the individual terms in their respective sums.) |
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Feb 26 |
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Is the Binomial Expectation of Convex Function Convex in p? Note that we can get a related lower bound using the technique in this answer: mathoverflow.net/questions/94083/…. Namely, wlog let $0 \leq p_1 \leq p_2 \leq 1$, $t \in [0,1]$ and $p = t p_1 + (1-t) p_2$. Then, if $Y = t X_1 + (1-t) X_2$ where $X_1 \sim \mathrm{Bin}(n, p_1)$ and $X_2 \sim \mathrm{Bin}(n,p_2)$, we have $g(p) \geq \mathbb E(t X_1 + (1-t) X_2)$. In other words, together with Noah's answer, $$\mathbb E h(t X_1 + (1-t) X_2) \leq \mathbb E h(X) \leq t \mathbb E h(X_1) + (1-t)\mathbb E h(X_2) \\,.$$ |
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Dec 17 |
awarded | ● Nice Answer |
