4
$\begingroup$

Here is the problem. I have to compute the limit:

$$ \lim_{q\to \infty} \left(\sum_{i=1}^m p_i^{q}\right)^{\frac{1}{1-q}} $$

where $p_i$s are numbers from $0$ to $1$ and $\sum_{i=1}^mp_i=1$. I found that solution should be $\frac1{p_{max}}$ but I don't know why.

Thanks a lot for your time.

$\endgroup$
  • $\begingroup$ We have that $$\Bigl(\sum_{i=1}^m p_i^q\Bigr)^{1/(1-q)}=\frac1{\|p\|_q\cdot\|p\|_q^{1/(q-1)}}$$ and $\|p\|_q\to\|p\|_\infty$ as $q\to\infty$, where $\|\cdot\|_q$ is the $\ell^q$ norm and $\|\cdot\|_\infty$ is the maximum norm. But I'm not sure how to conclude that the limit is actually $1/\|p\|_\infty$. $\endgroup$ – Cm7F7Bb Nov 11 '15 at 11:01
1
$\begingroup$

Use the inequality $$ p_{\text{max}}^q\le\sum_{i=1}^m p_i^{q}\le m\,p_{\text{max}}^q. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.