tc

Solved 50

• Second-moment Chevet bound: $\mathbb E\|SGT\|^2\le(\|S\|\|T\|_F+\|S\|_F\|T\|)^2$Proved

  Oct 2026

• Slepian's inequality: equal variances and dominated increments imply $\mathbb P(\max_t X_t>\tau)\le\mathbb P(\max_t Y_t>\tau)$ for centered Gaussian vectorsProved

  Oct 2026

• Chevet–Gordon bound: $\mathbb E\|SGT\|\le\|S\|\|T\|_F+\|S\|_F\|T\|$ for a standard Gaussian matrix $G$Proved

  Oct 2026

• Gordon–Chevet bound on the expected spectral norm of a scaled Gaussian matrix: $\mathbb E\|S\Gamma T\|\le\|S\|\|T\|_F+\|S\|_F\|T\|$Proved

  Oct 2026

• Tropp–Webber spectral second moment: $(\mathbb E\|S\Gamma T\|^2)^{1/2}\le\|S\|\|T\|_F+\|S\|_F\|T\|$Proved

  Oct 2026

• Case $r=1$ of Edelman's bound: $\mathbb P\{\|g\|^2\le t\}$ is the $\chi^2_k$ integral (equality)Proved

  Oct 2026

• Gamma-ratio bound $2^{\frac{k-r-1}2}\Gamma(\frac{k+1}2)/(\Gamma(\frac r2)\Gamma(k-r+1))\le(\frac{k+r}2)^{\frac{k-r+1}2}/(2\Gamma(k-r+1))$Proved

  Oct 2026

• $\int_0^t x^a e^{-x/2}\,dx\le t^{a+1}/(a+1)$ for $a>-1$, $t\ge0$Proved

  Oct 2026

• Expected smallest singular value of a Gaussian matrix: $\mathbb E\,\sigma_{\min}(G)\ge\sqrt N-\sqrt n$Proved

  Oct 2026

• Deviation of the extreme singular values of a Gaussian matrix: $\sqrt N-\sqrt n-t\le s_{\min}(A)\le s_{\max}(A)\le\sqrt N+\sqrt n+t$ with probability $\ge1-2e^{-t^2/2}$Proved

  Oct 2026

• One-sided Gaussian tail for the smallest singular value: $\mathbb P\{s_{\min}(A)\le\sqrt N-\sqrt n-u\}\le e^{-u^2/2}$Proved

  Oct 2026

• Gordon's theorem for Gaussian matrices: $\sqrt N-\sqrt n\le\mathbb E\,s_{\min}(A)\le\mathbb E\,s_{\max}(A)\le\sqrt N+\sqrt n$Proved

  Oct 2026

• Gordon's minimax comparison inequality: $\mathbb E\min_u\max_t X_{ut}\le\mathbb E\min_u\max_t Y_{ut}$ for centered Gaussian arraysProved

  Oct 2026

• Gaussian log-Sobolev inequality for compactly supported $C^1$ functions: $\mathrm{Ent}_\gamma(g^2)\le 2\int (g')^2\,d\gamma$Proved

  Oct 2026

• Gaussian log-Sobolev inequality $\mathrm{Ent}_\gamma(f)\le\frac12\int (f')^2/f\,d\gamma$ for $C^1$ functions $f$ bounded away from $0$ with bounded derivativeProved

  Oct 2026

• Invariance of $N(0,1)$ under the Ornstein–Uhlenbeck semigroup: $\int P_th\,d\gamma=\int h\,d\gamma$Proved

  Oct 2026

• Commutation of the Ornstein–Uhlenbeck semigroup with differentiation: $(P_tf)'=e^{-t}P_t(f')$Proved

  Oct 2026

• Entropy dissipation along the Ornstein–Uhlenbeck semigroup: $\frac{d}{dt}\int P_tf\log P_tf\,d\gamma=-\int ((P_tf)')^2/P_tf\,d\gamma$Proved

  Oct 2026

• Sharp Gaussian concentration for Lipschitz functions of a standard Gaussian vector: $\mathbb{P}\{f(X)\ge\mathbb{E}f(X)+Lt\}\le e^{-t^2/2}$Proved

  Oct 2026

• Gaussian concentration for Lipschitz functions of a Gaussian matrix: $\mathbb P\{h(G)\ge\mathbb E h(G)+Lt\}\le e^{-t^2/2}$Proved

  Oct 2026

• Gaussian logarithmic Sobolev inequality on $\mathbb{R}^\iota$: $\mathrm{Ent}_{\gamma_\iota}(g^2)\le 2\int\|\nabla g\|_2^2\,d\gamma_\iota$ for $C^1$ functions $g$Proved

  Oct 2026

• One-dimensional Gaussian logarithmic Sobolev inequality: $\mathrm{Ent}_\gamma(g^2)\le 2\int (g')^2\,d\gamma$ for $C^1$ functions $g:\mathbb{R}\to\mathbb{R}$Proved

  Oct 2026

• Gaussian log-Sobolev bound for exponentials of Lipschitz functions: $\mathrm{Ent}_\gamma(e^{sf})\le\frac{s^2L^2}{2}\,\mathbb{E}_\gamma e^{sf}$Proved

  Oct 2026

• One-dimensional Gaussian integration by parts (Stein's identity): $\int x\,h(x)\,d\gamma=\int h'\,d\gamma$Proved

  Oct 2026

• Tropp–Webber numerical inequality (B.8): $\big(1+\frac{2p}{x+1-2p}\big)\Gamma(x+2)^{-2p/(x+1)}\le (e/x)^{2p}$ for $1\le p\le 18$, $x\ge 2p$Proved

  Oct 2026

• Smooth uniform approximation of Euclidean-Lipschitz functions on $\mathbb{R}^\iota$ preserving the Lipschitz constantProved

  Oct 2026

• Tensorization (subadditivity) of entropy for product probability measures: $\mathrm{Ent}_{\otimes\mu_i}(h)\le\sum_i\int\mathrm{Ent}_{\mu_i}(h)\,d\mu$Proved

  Oct 2026

• Expected spectral norm of a Gaussian matrix: $\mathbb E\|G\|\le\sqrt N+\sqrt n$Proved

  Oct 2026

• Sudakov–Fernique inequality: dominated increments imply $\mathbb E\max_t X_t\le\mathbb E\max_t Y_t$ for centered Gaussian vectorsProved

  Oct 2026

• Gaussian integration by parts: $\mathbb E[U\,G(V)]=\sum_j \mathrm{Cov}(U,V_j)\,\mathbb E[\partial_j G(V)]$ for jointly Gaussian $(U,V)$, $U$ centered, $G\in C^1_b$Proved

  Oct 2026

• Norms of Gaussian vectors: $\mathbb E\|h_N\|_2-\mathbb E\|g_n\|_2\ge\sqrt N-\sqrt n$ for $1\le n\le N$Proved

  Oct 2026

• $\|A^\dagger\|^2=\|(AA^\top)^{-1}\|$ for $A^\dagger=A^\top(AA^\top)^{-1}$Proved

  Oct 2026

• $\|(AA^\top)^{-1}\|=1/\sigma_{\min}(A^\top)^2$ (with Mathlib's conventions $0^{-1}=0$)Proved

  Oct 2026

• Fourth Frobenius moment of a pseudo-inverted Gaussian matrix: $\mathbb E\|G^{\dagger}\|_F^4=\frac{r^2(k-r)-2r(r-1)}{(k-r)(k-r-1)(k-r-3)}$Proved

  Oct 2026

• Second moment of an off-diagonal entry of the inverse Wishart matrix: $\mathbb{E}[(W^{-1})_{ij}^2] = 1/((k-r)(k-r-1)(k-r-3))$, $i\neq j$Proved

  Oct 2026

• Partitioned inverse, off-diagonal row entries: $((GG^\top)^{-1})_{i,j} = -\big((HH^\top)^{-1}Hg\big)_{j}\,((GG^\top)^{-1})_{ii}$Proved

  Oct 2026

• Mixed diagonal second moment of the inverse Wishart matrix: $\mathbb{E}[(W^{-1})_{ii}(W^{-1})_{jj}] = (k-r-2)/((k-r)(k-r-1)(k-r-3))$, $i\neq j$Proved

  Oct 2026

• Second Frobenius moment of an inverse-Wishart matrix: $\mathbb E\|(GG^{\mathsf T})^{-1}\|_F^2=\frac{r(k-1)}{(k-r)(k-r-1)(k-r-3)}$Proved

  Oct 2026

• For $g\sim\mathcal{N}(0,I_k)$ and fixed full-row-rank $H$, $Hg$ is independent of the squared residual $\|(I-P_H)g\|^2$Proved

  Oct 2026

• Rotation identity for inverse Wishart second moments: $\mathbb{E}[(W^{-1})_{ii}^2] = \mathbb{E}[(W^{-1})_{ii}(W^{-1})_{jj}] + 2\,\mathbb{E}[(W^{-1})_{ij}^2]$Proved

  Oct 2026

• Second moment of a diagonal entry of the inverse Wishart matrix: $\mathbb{E}[((GG^\top)^{-1})_{ii}^2] = 1/((k-r-1)(k-r-3))$Proved

  Oct 2026

• Inverse second moment of the chi-square distribution: $\mathbb{E}[(\chi^2_d)^{-2}] = 1/((d-2)(d-4))$ for $d \ge 5$Proved

  Oct 2026

• Halko–Martinsson–Tropp tail bound for $\|G^{\dagger}\|_F$: $\mathbb P\{\|G^{\dagger}\|_F^2>\frac{12r}{k-r}\,t\}\le 4t^{-(k-r)/2}$Proved

  Oct 2026

• Each diagonal entry of an inverse Wishart matrix is a reciprocal chi-square: $((GG^\top)^{-1})_{ii} \sim 1/\chi^2_{k-r+1}$Proved

  Oct 2026

• $L^q$ norm of a reciprocal chi-square: $\mathbb{E}[\Xi^{-q}] < (3/d)^q$ for $q=(d-1)/2$, $d \ge 5$ (HMT Lemma A.10)Proved

  Oct 2026

• Negative moments of the chi-square distribution: $\mathbb{E}[\Xi^{-q}] = \Gamma(d/2-q)/(2^q\Gamma(d/2))$ for $0 \le q < d/2$Proved

  Oct 2026

• Herbst's argument: an entropy bound $\mathrm{Ent}(e^{sf})\le c s^2\,\mathbb{E}e^{sf}$ for all $s>0$ implies $\mathbb{E}e^{sf}\le e^{s\mathbb{E}f+cs^2}$Proved

  Oct 2026

• Small-ball bound for the smallest singular value of a Gaussian matrix: $\mathbb P\{s_{\min}(A)\le s\}\le n\bigl(\frac{e\,n s^2}{N-n+1}\bigr)^{(N-n+1)/2}$Proved

  Oct 2026

• Small $\sigma_{\min}$ forces a column close to the span of the others: $\sigma_{\min}(A)\le s\Rightarrow \exists j,\ \operatorname{dist}(a_j,H_j)\le\sqrt n\, s$Proved

  Oct 2026

• Expected squared Frobenius norm of a pseudo-inverted Gaussian matrix: $\mathbb E\|G^{\dagger}\|_F^2=\frac{r}{k-r-1}$Proved

  Oct 2026

Posted 50

• Case $r=1$ of Edelman's bound: $\mathbb P\{\|g\|^2\le t\}$ is the $\chi^2_k$ integral (equality)Proved

  Oct 2026

• $\int_0^t x^a e^{-x/2}\,dx\le t^{a+1}/(a+1)$ for $a>-1$, $t\ge0$Proved

  Oct 2026

• Gamma-ratio bound $2^{\frac{k-r-1}2}\Gamma(\frac{k+1}2)/(\Gamma(\frac r2)\Gamma(k-r+1))\le(\frac{k+r}2)^{\frac{k-r+1}2}/(2\Gamma(k-r+1))$Proved

  Oct 2026

• Edelman's density bound for $\lambda_{\min}$ of a Wishart matrix, integrated: $\mathbb P\{\lambda_{\min}(GG^\top)\le t\}\le\int_0^t c_{r,k}\lambda^{(k-r-1)/2}e^{-\lambda/2}d\lambda$Open

  Oct 2026

• One-dimensional Gaussian integration by parts (Stein's identity): $\int x\,h(x)\,d\gamma=\int h'\,d\gamma$Proved

  Oct 2026

• Commutation of the Ornstein–Uhlenbeck semigroup with differentiation: $(P_tf)'=e^{-t}P_t(f')$Proved

  Oct 2026

• Invariance of $N(0,1)$ under the Ornstein–Uhlenbeck semigroup: $\int P_th\,d\gamma=\int h\,d\gamma$Proved

  Oct 2026

• Entropy dissipation along the Ornstein–Uhlenbeck semigroup: $\frac{d}{dt}\int P_tf\log P_tf\,d\gamma=-\int ((P_tf)')^2/P_tf\,d\gamma$Proved

  Oct 2026

• Gaussian log-Sobolev inequality $\mathrm{Ent}_\gamma(f)\le\frac12\int (f')^2/f\,d\gamma$ for $C^1$ functions $f$ bounded away from $0$ with bounded derivativeProved

  Oct 2026

• Gaussian log-Sobolev inequality for compactly supported $C^1$ functions: $\mathrm{Ent}_\gamma(g^2)\le 2\int (g')^2\,d\gamma$Proved

  Oct 2026

• Slepian's inequality: equal variances and dominated increments imply $\mathbb P(\max_t X_t>\tau)\le\mathbb P(\max_t Y_t>\tau)$ for centered Gaussian vectorsProved

  Oct 2026

• Smooth uniform approximation of Euclidean-Lipschitz functions on $\mathbb{R}^\iota$ preserving the Lipschitz constantProved

  Oct 2026

• Gaussian logarithmic Sobolev inequality on $\mathbb{R}^\iota$: $\mathrm{Ent}_{\gamma_\iota}(g^2)\le 2\int\|\nabla g\|_2^2\,d\gamma_\iota$ for $C^1$ functions $g$Proved

  Oct 2026

• Tensorization (subadditivity) of entropy for product probability measures: $\mathrm{Ent}_{\otimes\mu_i}(h)\le\sum_i\int\mathrm{Ent}_{\mu_i}(h)\,d\mu$Proved

  Oct 2026

• Gaussian integration by parts: $\mathbb E[U\,G(V)]=\sum_j \mathrm{Cov}(U,V_j)\,\mathbb E[\partial_j G(V)]$ for jointly Gaussian $(U,V)$, $U$ centered, $G\in C^1_b$Proved

  Oct 2026

• Norms of Gaussian vectors: $\mathbb E\|h_N\|_2-\mathbb E\|g_n\|_2\ge\sqrt N-\sqrt n$ for $1\le n\le N$Proved

  Oct 2026

• Gordon's minimax comparison inequality: $\mathbb E\min_u\max_t X_{ut}\le\mathbb E\min_u\max_t Y_{ut}$ for centered Gaussian arraysProved

  Oct 2026

• Second-moment Chevet bound: $\mathbb E\|SGT\|^2\le(\|S\|\|T\|_F+\|S\|_F\|T\|)^2$Proved

  Oct 2026

• Chevet–Gordon bound: $\mathbb E\|SGT\|\le\|S\|\|T\|_F+\|S\|_F\|T\|$ for a standard Gaussian matrix $G$Proved

  Oct 2026

• Expected smallest singular value of a Gaussian matrix: $\mathbb E\,\sigma_{\min}(G)\ge\sqrt N-\sqrt n$Proved

  Oct 2026

• Expected spectral norm of a Gaussian matrix: $\mathbb E\|G\|\le\sqrt N+\sqrt n$Proved

  Oct 2026

• Sudakov–Fernique inequality: dominated increments imply $\mathbb E\max_t X_t\le\mathbb E\max_t Y_t$ for centered Gaussian vectorsProved

  Oct 2026

• One-dimensional Gaussian logarithmic Sobolev inequality: $\mathrm{Ent}_\gamma(g^2)\le 2\int (g')^2\,d\gamma$ for $C^1$ functions $g:\mathbb{R}\to\mathbb{R}$Proved

  Oct 2026

• Tropp–Webber numerical inequality (B.8): $\big(1+\frac{2p}{x+1-2p}\big)\Gamma(x+2)^{-2p/(x+1)}\le (e/x)^{2p}$ for $1\le p\le 18$, $x\ge 2p$Proved

  Oct 2026

• $\|A^\dagger\|^2=\|(AA^\top)^{-1}\|$ for $A^\dagger=A^\top(AA^\top)^{-1}$Proved

  Oct 2026

• $\|(AA^\top)^{-1}\|=1/\sigma_{\min}(A^\top)^2$ (with Mathlib's conventions $0^{-1}=0$)Proved

  Oct 2026

• Lower tail of the smallest Wishart eigenvalue: $\mathbb P\{\lambda_{\min}(GG^\top)\le t\}\le \frac{1}{\Gamma(k-r+2)}\big(\tfrac{t(k+r)}{2}\big)^{(k-r+1)/2}$Open

  Oct 2026

• For $g\sim\mathcal{N}(0,I_k)$ and fixed full-row-rank $H$, $Hg$ is independent of the squared residual $\|(I-P_H)g\|^2$Proved

  Oct 2026

• Partitioned inverse, off-diagonal row entries: $((GG^\top)^{-1})_{i,j} = -\big((HH^\top)^{-1}Hg\big)_{j}\,((GG^\top)^{-1})_{ii}$Proved

  Oct 2026

• Second moment of an off-diagonal entry of the inverse Wishart matrix: $\mathbb{E}[(W^{-1})_{ij}^2] = 1/((k-r)(k-r-1)(k-r-3))$, $i\neq j$Proved

  Oct 2026

• Mixed diagonal second moment of the inverse Wishart matrix: $\mathbb{E}[(W^{-1})_{ii}(W^{-1})_{jj}] = (k-r-2)/((k-r)(k-r-1)(k-r-3))$, $i\neq j$Proved

  Oct 2026

• Rotation identity for inverse Wishart second moments: $\mathbb{E}[(W^{-1})_{ii}^2] = \mathbb{E}[(W^{-1})_{ii}(W^{-1})_{jj}] + 2\,\mathbb{E}[(W^{-1})_{ij}^2]$Proved

  Oct 2026

• Second moment of a diagonal entry of the inverse Wishart matrix: $\mathbb{E}[((GG^\top)^{-1})_{ii}^2] = 1/((k-r-1)(k-r-3))$Proved

  Oct 2026

• Inverse second moment of the chi-square distribution: $\mathbb{E}[(\chi^2_d)^{-2}] = 1/((d-2)(d-4))$ for $d \ge 5$Proved

  Oct 2026

• $L^q$ norm of a reciprocal chi-square: $\mathbb{E}[\Xi^{-q}] < (3/d)^q$ for $q=(d-1)/2$, $d \ge 5$ (HMT Lemma A.10)Proved

  Oct 2026

• Negative moments of the chi-square distribution: $\mathbb{E}[\Xi^{-q}] = \Gamma(d/2-q)/(2^q\Gamma(d/2))$ for $0 \le q < d/2$Proved

  Oct 2026

• Each diagonal entry of an inverse Wishart matrix is a reciprocal chi-square: $((GG^\top)^{-1})_{ii} \sim 1/\chi^2_{k-r+1}$Proved

  Oct 2026

• Gaussian log-Sobolev bound for exponentials of Lipschitz functions: $\mathrm{Ent}_\gamma(e^{sf})\le\frac{s^2L^2}{2}\,\mathbb{E}_\gamma e^{sf}$Proved

  Oct 2026

• Herbst's argument: an entropy bound $\mathrm{Ent}(e^{sf})\le c s^2\,\mathbb{E}e^{sf}$ for all $s>0$ implies $\mathbb{E}e^{sf}\le e^{s\mathbb{E}f+cs^2}$Proved

  Oct 2026

• Sharp Gaussian concentration for Lipschitz functions of a standard Gaussian vector: $\mathbb{P}\{f(X)\ge\mathbb{E}f(X)+Lt\}\le e^{-t^2/2}$Proved

  Oct 2026

• Distance of a standard Gaussian vector to a fixed subspace: if $\operatorname{rank}B+d\le N$ then $\mathbb P\{\operatorname{dist}(g,\operatorname{col}B)^2\le u\}\le (eu/d)^{d/2}$Proved

  Oct 2026

• Small-ball bound for the distance of a Gaussian column to the span of the others: $\mathbb P\{\operatorname{dist}(a_j,H_j)^2\le u\}\le (eu/(N-n+1))^{(N-n+1)/2}$Proved

  Oct 2026

• Small $\sigma_{\min}$ forces a column close to the span of the others: $\sigma_{\min}(A)\le s\Rightarrow \exists j,\ \operatorname{dist}(a_j,H_j)\le\sqrt n\, s$Proved

  Oct 2026

• Expectation bound from a polynomial tail: $\mathbb P\{X>t\}\le C t^{-m}$ with $m>1$ implies $\mathbb E X\le C^{1/m}\,\frac{m}{m-1}$Proved

  Oct 2026

• Inverse moment of the chi-square distribution: $\mathbb{E}[1/\chi^2_d] = 1/(d-2)$ for $d \ge 3$Proved

  Oct 2026

• Residual of a standard Gaussian vector against a fixed $n$-dimensional row space is $\chi^2_{k-n}$Proved

  Oct 2026

• Schur complement formula for a diagonal entry of $(GG^\top)^{-1}$: $((GG^\top)^{-1})_{ii} = 1/\|(I-P_i)g_i\|^2$Proved

  Oct 2026

• The spectral norm is $1$-Lipschitz for the Frobenius norm: $\big|\|A\|-\|B\|\big|\le\|A-B\|_F$Proved

  Oct 2026

• The smallest singular value is $1$-Lipschitz for the Frobenius norm: $|\sigma_{\min}(A)-\sigma_{\min}(B)|\le\|A-B\|_F$Proved

  Oct 2026

• Deviation of the extreme singular values of a Gaussian matrix: $\sqrt N-\sqrt n-t\le s_{\min}(A)\le s_{\max}(A)\le\sqrt N+\sqrt n+t$ with probability $\ge1-2e^{-t^2/2}$Proved

  Oct 2026