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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?