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Assumptions such as smoothness will not guarantee this. Consider $f(x,y)=\varphi((1+y^2)x)$ with a $\varphi\in C^{\infty}(\mathbb R)$ with $\varphi> 0$, $\varphi(0)=1$, $\int\varphi = 1/\pi$. Then $f\ …

Anthony's comment has the correct answer: $Z=(z_1,\ldots ,z_N)$ is uniformly distributed on $\|Z\|^2=N$, $\langle e, Z\rangle=0$, $e=(1,1,\ldots, 1)$ (that is, the joint distribution is the $(N-2)$-di …

The best constant is $C^{d-1}\sqrt{d}$. Write $D(A)=\det A$. We can rephrase the inequality as the claim that $\|D'\|_F\le L$ for $c\le A\le C$. (It's perhaps best to think of the matrices as long col …

We can try the method from the second answer to the first question you linked to. I originally thought that would clarify everything, but that was based on a miscalculation. I can now finally prove so …

As soon as $\phi(x)$ decays too rapidly, you are doomed. For example, if $\phi(x)=e^{-x^2}$, then $\psi(x)=Ee^{itY}$ would have to satisfy $|\psi(x)|=e^{-x^2/2}$, but now the rapid decay will make the …

My initial intuition was incorrect. It seems we can sometimes solve this explicitly, and the minimizer does not have full support; my argument is incomplete (as discussed below), but I think it's reas …

This will not work in general. Let's take $a$ as a vector with iid $\pm 1$ entries, with probability $1/2$ for each of $\pm 1$, and $x=(1,1,\ldots ,1)$ (normalize this if you prefer). Then if $\langle …

This is an expanded version of my comment above. The probability equals $1$ if we make the mild extra assumption that $t\le 1$ and $$ t\le \frac{c-1}{c+1}\left(\frac{1}{p} - 1\right) \quad\quad\quad\q …

Let's write $p_n(x)=P(S_n\le nx)$; we want to show that this increases in $n$ for fixed $x>1/2$. By conditioning on $X_{n+1}$, we obtain that $$ p_{n+1}(x) = \int_0^1 p_n\left( x+ \frac{x-y}{n} \right …

For any finite sequence $c_1, \ldots , c_d$ which can be a moment sequence at all (which is characterized by a positive definiteness condition), there is a description of all probability measures $\mu …

As already pointed out by Anthony in a comment, you can't really expect explicit formulae. Let's write $p(x)$ for the probability that the RW starting at $x\in (a,b)$ hits $a$ before it hits $b$. Then …

Yes, if $\tau$ is defined as $\min \{t\ge 0: Y_t=0 \}$. Condition on $\tau =t$ and use that $Ee^{i\theta X_t}=e^{i\theta x-\theta^2 t/2}$, as explained in the answer to the linked question. Thus $$ Ee …

The argument from the paper Carlo linked to (which, by the way, is essentially a classical result of Cramer's) can be adapted to your situation. Consider the moment generating function $f(z) = Ee^{z\ …