Let \(\mu \) be a probability measure on a measurable space \(\Omega \), let \(\omega \mapsto A(\omega )\) be a family of \(k \times d\) real matrices indexed by \(\Omega \), and let \(\varepsilon \in \mathbb {R}\). Then the probability that the zero vector \(0 \in \mathbb {R}^d\) triggers the bad distortion event, namely that

satisfies

Let \(X\colon \Omega \to \mathbb {R}\) be a measurable random variable on a measure space \((\Omega , \mu )\) whose law is the centered Gaussian \(N(0, \sigma _0)\) with variance \(\sigma _0 \ge 0\). Then for every scalar \(c \in \mathbb {R}\), the scaled variable \(c\, X\) has law

the centered Gaussian with variance \(c^2 \sigma _0\).

Let \((\Omega ,\mu )\) be a probability space, and let \((Y_j)_{j}\) be a family of independent, measurable real-valued random variables, each distributed as a centred Gaussian with mean \(0\) and variance \(\sigma _0 \ge 0\). Then for any real coefficients \((x_j)_j\) and any finite index set \(s\), the random variable \(\sum _{j \in s} x_j\, Y_j\) has distribution \(N\! \Bigl(0,\; \bigl(\sum _{j \in s} x_j^2\bigr)\, \sigma _0\Bigr)\); that is, its law is the centred Gaussian on \(\mathbb {R}\) with variance \(\sigma _0 \sum _{j \in s} x_j^2\).

Let \((\Omega ,\mu )\) be a probability space, and let \(A\) be a random \(k\times d\) real matrix, that is, a measurable map assigning to each \(\omega \in \Omega \) a matrix \(A(\omega )\in \mathbb {R}^{k\times d}\). Suppose the rows of \(A\) are mutually independent, meaning the \(k\) random vectors \(\omega \mapsto \bigl(A(\omega )_{i,1},\dots ,A(\omega )_{i,d}\bigr)\in \mathbb {R}^{d}\), for \(i=1,\dots ,k\), are mutually independent under \(\mu \). Then for every fixed \(x\in \mathbb {R}^{d}\) the \(k\) coordinates of the product \(A(\omega )\, x\), namely the real-valued random variables \(\omega \mapsto (A(\omega )\, x)_i=\sum _{j=1}^{d}A(\omega )_{i,j}\, x_j\) for \(i=1,\dots ,k\), are mutually independent under \(\mu \).

Let \(A\in \mathbb {R}^{k\times d}\) be a \(k\times d\) real matrix, regarded as the linear map from the Euclidean space \(\mathbb {R}^{d}\) to \(\mathbb {R}^{k}\) that it induces, and let \(x\in \mathbb {R}^{d}\). Then for each index \(i\in \{ 1,\dots ,k\} \), the \(i\)-th coordinate of the image \(Ax\) is given by

Let \(A\) be a \(k \times d\) real matrix, regarded as the linear map \(x \mapsto Ax\) from the Euclidean space \(\mathbb {R}^d\) to \(\mathbb {R}^k\), and let \(x \in \mathbb {R}^d\). Then the squared Euclidean norm of the image is the sum of the squares of its coordinates,

Let \(h \colon \mathbb {R}\to \mathbb {R}\) be the function \(h(x) = x^2 + x + \log (1 - x)\). Then at every point \(u \lt 1\), the function \(h\) is differentiable with derivative

The real-valued expression \(x^2 + x + \log (1 - x)\) takes the value \(0\) at \(x = 0\); that is, \(0^2 + 0 + \log (1 - 0) = 0\).

For every real number \(s\) with \(\left\lvert s\right\rvert \le 1/4\), one has

Let \(Z\) be a standard Gaussian random variable, \(Z \sim N(0,1)\), and let \(s \in \mathbb {R}\) satisfy \(2s \lt 1\). Then

Let \(v \gt 0\) and write \(N(0,v)\) for the centered Gaussian distribution on \(\mathbb {R}\) with mean \(0\) and variance \(v\). Then for every real number \(t\) with \(2tv \lt 1\),

Let \(v\) be a nonnegative real number with \(v \ne 0\), and let \(N(0,v)\) denote the centred Gaussian probability measure on \(\mathbb {R}\) with variance \(v\). If \(t \in \mathbb {R}\) satisfies \(2tv \lt 1\), then the function \(y \mapsto e^{t y^2}\) is integrable with respect to \(N(0,v)\).

Let \(\mu \) be a probability measure on \(\Omega \), let \(k \ge 1\), and let \(Y_1,\dots ,Y_k : \Omega \to \mathbb {R}\) be independent random variables, each with law \(N(0, 1/k)\). Then for every \(\varepsilon \) with \(0 \lt \varepsilon \lt 1\),