# Tag Info

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Complementing the answer by Carlo, if you take all $k$'s equal you have $$f_{\rm GUE(d)}(k,...,k)\propto \int dX e^{ik{\rm Tr}(X)}e^{-\frac{d}{2}{\rm Tr}(X^2)}.$$ Taking $x$ to be any real diagonal element from $X$, this is $$f_{\rm GUE(d)}(k,...,k)\propto \left(\int dx e^{ikx}e^{-\frac{d}{2}x^2}\right)^d.$$ I think in the end you have simply $f_{\rm GUE(d)}(... 4 The Fourier transform of the marginal distribution of a single eigenvalue in the GUE is known, $$f_{{\rm GUE}(d)}(k,0,0,\ldots,0)=e^{-\tfrac{1}{2}k^2/d}\sum_{j=0}^{d-1}(-1)^jk^{2j}\frac{(d-1)(d-2)\cdots(d-j)}{j!(j+1)!d^j},$$ see these lecture notes. A curiosity: the$d=2$result is given in this publication in terms of the confluent hypergeometric function$...

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The classical theorem of Darmois-Skitovich says that two linear forms (with all non-zero coefficients, and of at least two variables) of independent random variables are independent only if the random variables are normal. Combined with the comment of @Gerald Edgar, this completely settles the question. Ref. A. Kagan, Yu. Linnik, C. Rao, Characterization ...

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If the cardinality of the set $S_f$ is $\ge1$, then the value of the integral $$I:=\int_0^1\Big(\frac{f'(x)}{f(x)}\bigg)^2 dx$$ will be $\infty$ with nonzero probability, because with nonzero probability the polynomial $f$ will have a non-multiple root in the interval $[0,1]$. This will then also imply that the expectation of $I$ is $\infty$.

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We prove the weaker bound $$\mathbf{P} \left[ \|O \mathbb{x}\|_1 \leq \frac{cd}{\sqrt{\log d}} \right] \leq 2^{-Cd}$$ for some constants $C, c$. Define the Gaussian mean width of a compact subset $A \subset \mathbf{R}^d$ as $$w(A) = \mathbf{E} \sup_{x \in A} \langle G,x \rangle$$ where $G$ is a standard Gaussian vector in $\mathbf{R}^d$. We use the ...

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Here is an attempt to the problem for a worst-case $O$, with worse constants. So fix $O$, letting $o_i$ denote its $i$th row, and take $X$ random in $\{0,1\}^d$. We claim that $E |\langle o_i, X\rangle| \ge cst$. To see this, write $$\langle o_i, X\rangle = \langle o_i, \frac{{\bf 1}}{2}\rangle + \langle o_i, (X - \frac{{\bf 1}}{2})\rangle$$ and assume ...

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