Timeline for Binomial series
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 10 at 22:35 | comment | added | Iosif Pinelis | I have now added the probabilistic proof of the uniform convergence. | |
| Jan 10 at 21:51 | comment | added | Giorgio Metafune | @IosifPinelis Yes true. I wonder if your proof above, based on dominated convegence, yields only pointwise convergence. | |
| Jan 10 at 21:45 | comment | added | Iosif Pinelis | Yes, the probabilistic proof of the uniform convergence for continuous $f$ follows from two facts: (i) $f$ is uniformly continuous on the compact set $[0,1]$ and (ii) the variance of the binomial distribution with parameters $n,p$ is $p(1-p)/n\le1/(4n)\to0$ as $n\to\infty$ uniformly in $p\in[0,1]$. At least in this sense, the binomial distribution with parameters $n,p$ is the least concentrated (near its mean) when $p=1/2$ (the entropy is then also the largest). | |
| Jan 10 at 21:39 | comment | added | Giorgio Metafune | @IosifPinelis I know this for Bernstein polynomials. Do you mean that your proof shows uniform convergence? | |
| Jan 10 at 21:32 | comment | added | Iosif Pinelis | The convergence is uniform for continuous $f$. | |
| Jan 10 at 21:11 | comment | added | Pietro Majer | Azz, you beat me for 1 minute! :) | |
| Jan 10 at 21:06 | history | answered | Giorgio Metafune | CC BY-SA 4.0 |