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 Apr 28 awarded Tumbleweed Apr 21 asked Equivalent conditions for weak $L^p$ spaces for $p\leq 1$ Apr 15 accepted Convergence of a series of functions almost everywhere Apr 15 comment Monotonicity property almost everywhere Thanks I think I know how to do this. Apr 10 comment Convergence of a series of functions almost everywhere @DavidC.Ullrich I checked the problem and it's as it is. If $f$ is the characteristic function of a compact interval as you suggested, wouldn't the sum then becomes just a finite sum for any $x$, and so will converge ? (since $x-\sqrt{n}$ would eventually leave the support of this characteristic function for large enough $n$). The problem might be false, but is there an explicit counterexample to this? Apr 10 comment Convergence of a series of functions almost everywhere @LeGrandDODOM Yes I checked the problem and it is $n^{-1/2}$. It's true that $\int \sum_{n=1}^\infty |n^{-1/2} f(x-\sqrt{n})|= +\infty$, but that doesn't imply that the sum is equal to $+\infty$ for $x$ in a set of positive measure, right? Apr 10 revised Convergence of a series of functions almost everywhere edited tags Apr 10 asked Monotonicity property almost everywhere Apr 10 asked Convergence of a series of functions almost everywhere Feb 8 awarded Revival Nov 5 awarded Popular Question Jul 7 awarded Yearling Apr 16 comment Entropy upper bound inequality for Sub-Gaussian Random Variable But that is a useless inequality because it's easy to see that $\text{Ent}(Z)\geq 0$ for any non-negative random variable $Z$. Apr 13 answered For all $\epsilon>0$ there exists $f,g$ such that $\|f\ast g\|_p>(1-\epsilon)\|f\|_1\|g\|_p$. Apr 13 awarded Revival Apr 13 answered Operator norm of a convolution Apr 13 comment If $R$ and $s$ are the range and standard deviation of a set of $n$ values, then $4\le R^2/s^2\le 2n$ @futurebird Translating the data (adding all the data by the same constant) does not affect both variance and range. Say if the data is from $a$ to $a+R$, we could add $-a$ to all of them so that $0\leq x_i\leq R$ and the variance and the range will stay the same. This will not change the result. Apr 13 revised If $R$ and $s$ are the range and standard deviation of a set of $n$ values, then $4\le R^2/s^2\le 2n$ deleted 73 characters in body Apr 13 awarded Commentator Apr 13 comment Entropy upper bound inequality for Sub-Gaussian Random Variable I'm still not sure with this part: You said in the beginning that we have $$\frac{\log M(t)}{t} \leq \frac{t\sigma^2}{2}+ tE[X].$$If I understand correctly, after doing the Taylor expansion, you truncate it to get that the first two terms in the expansion is less than $\frac{t\sigma^2}{2}+ tE[X]$. It seems that you are assuming the remainder part is always non-negative for all $t$. Why is this true? Or did I misunderstand some supposedly simple argument?