A quadratic form is the expression $X^\top A X = \sum_{i,j} A_{ij} X_i X_j$ for a fixed $n \times n$ matrix $A$ and a random vector $X = (X_1, \dots, X_n)$ with independent, mean-zero entries. The squared length $\|X\|^2$ is the case $A = I$, a sample variance is a quadratic form, and so is the energy $\|M x\|^2 = x^\top (M^\top M) x$ that a fixed linear map $M$ measures. The question is how tightly $X^\top A X$ concentrates around its mean. The Hanson–Wright inequality is the standard answer, and the answer is not a single Gaussian tail but two regimes.

The mean is the trace

Take the entries to be independent with mean zero and variance $1$. Then $\mathbb{E}[X_i X_j] = \delta_{ij}$, equal to $1$ when $i = j$ and $0$ otherwise, so only the diagonal of $A$ survives the expectation:

So $X^\top A X$ fluctuates around $\operatorname{tr}(A)$.

The Gaussian case, by rotating to the eigenbasis

Take $X \sim \mathcal{N}(0, I_n)$ and $A$ symmetric. (Any quadratic form sees only the symmetric part $\tfrac{1}{2}(A + A^\top)$, since $x^\top A x = x^\top A^\top x$, so assuming symmetry loses nothing.) Diagonalize $A = Q \Lambda Q^\top$ with orthonormal $Q$ and eigenvalues $\lambda_1, \dots, \lambda_n$. The rotated vector $g = Q^\top X$ is again standard Gaussian, since the standard Gaussian is rotation invariant, and in these coordinates the quadratic form is a plain weighted sum of squares:

Subtracting the mean $\operatorname{tr}(A) = \sum_i \lambda_i$,

a sum of independent terms, one per eigenvalue.

Each term $g_i^2 - 1$ is a centered chi-square with one degree of freedom. It is sub-exponential, meaning its tail decays like $e^{-t/2}$ rather than a Gaussian’s $e^{-t^2/2}$, because squaring a Gaussian fattens the tail. Its moment generating function is

Scaling the $i$-th term by $\lambda_i$ and using independence, the cumulant generating function of the whole sum is at most

where $\|A\|_F^2 = \sum_i \lambda_i^2$ is the squared Frobenius norm and the band $|s| \le \tfrac{1}{4\|A\|}$ comes from needing $|s\lambda_i| \le \tfrac{1}{4}$ for every $i$, with $\|A\| = \max_i |\lambda_i|$ the operator norm (the largest singular value).

This is the cumulant bound of a Bernstein-type variable: a Gaussian-like $s^2$ term, but valid only inside a band whose width is set by $\|A\|$. Optimizing the Chernoff bound $\Pr[S \ge t] \le \exp(-st + 2 s^2 \|A\|_F^2)$ over $s$ in that band gives the two regimes. For small $t$ the optimal $s$ stays inside the band and the bound is Gaussian; for large $t$ the optimum is pinned at the edge $s = \tfrac{1}{4\|A\|}$ and the bound is exponential. Together,

That is the Hanson–Wright inequality for Gaussian inputs.

For $A = I$, $X^\top A X = \|X\|^2$, the mean is $\operatorname{tr}(I) = n$, and the norms are $\|I\|_F^2 = n$ and $\|I\| = 1$, so

Fluctuations of size $t \lesssim n$ are Gaussian; past $t \approx n$ the tail turns exponential. This is the concentration of $\|X\|^2$ behind the thin shell in high-dimensional Gaussians: the squared length sits at $n$ with fluctuations of order $\sqrt{n}$, so the length sits at $\sqrt{n}$ with fluctuations of order $1$.

Two norms, two regimes

The Frobenius norm $\|A\|_F^2 = \sum_i \lambda_i^2$ is the total variance: $\operatorname{Var}(X^\top A X) = 2\sum_i \lambda_i^2 = 2\|A\|_F^2$ for Gaussian $X$. It sets the Gaussian regime near the mean, where the fluctuation is an average over all $n$ eigenvalue-terms and the central-limit effect applies.

The operator norm $\|A\| = \max_i |\lambda_i|$ is the single largest weight. The heaviest-tailed term in $\sum_i \lambda_i (g_i^2 - 1)$ is the one with the biggest $|\lambda_i|$, and it alone carries a sub-exponential tail $e^{-t/(\text{const} \cdot \|A\|)}$ that no averaging removes. It sets the far tail.

The crossover between the two sits at $t^\star = \|A\|_F^2 / \|A\|$. Below it the form looks Gaussian; above it a single dominant direction takes over and the tail is exponential. The curve below is the rate $\min(t^2/\|A\|_F^2,\; t/\|A\|)$ that sits in the exponent: the larger the rate, the smaller the tail. The parabola governs small deviations, the line governs large ones, and they switch at $t^\star$. Slide the two norms and watch the crossover move.

What decides which regime matters in practice is the shape of the spectrum. Below, the bars are the eigenvalues $\lambda_i$, normalized so the largest is $\|A\| = 1$. The slider tilts the spectrum from flat, where every direction counts equally, like $A = I$, to spiky, where one direction dominates. A flat spectrum pushes the crossover $t^\star$ far out, so the form is Gaussian over a wide range; a spiky spectrum pulls it in, so the exponential tail takes over early. The ratio $\|A\|_F^2 / \|A\|^2$ that sets the crossover (in units of $\|A\|$) is the stable rank, an effective count of active directions.

The general theorem

For general independent entries the rotation that handled the Gaussian case is not available: only the Gaussian is rotation invariant, so for any other distribution the rotated coordinates $Q^\top X$, while still sub-Gaussian, are no longer independent, and the clean weighted sum of squares falls apart. The inequality holds anyway. For a random vector with independent, mean-zero, sub-Gaussian entries of sub-Gaussian norm at most $K$ (the scale on which the entries’ own tails decay like $e^{-x^2/K^2}$),

the same two-regime shape, with $K$ tracking how heavy the entries are.

The general proof splits the form along its diagonal,

The diagonal part is a sum of independent variables $A_{ii} X_i^2$; it carries the mean $\operatorname{tr}(A)$ and concentrates by Bernstein’s inequality (the same sub-exponential bound used above, now for a sum rather than after a rotation), no harder than the Gaussian case. The off-diagonal part has mean zero and is the real content of the theorem. Its terms $X_i X_j$ share variables, so they are not independent, and the standard handle is decoupling: replace one copy of $X$ by an independent copy $X'$, so that conditionally on $X'$ the off-diagonal sum $\sum_{i\ne j} A_{ij} X_i X_j'$ is a linear form in the independent variables $X_i$, which sub-Gaussian tools control directly. Carrying that through reproduces the same two norms.

Where it gets used

The inequality is a standard tool wherever a squared length or an energy has to be pinned to its mean.

• Length and distance concentration. $\|X\|^2$ around $n$ is the case $A = I$. More generally $\|Mx\|^2 = x^\top(M^\top M)x$ concentrates around its mean for a fixed map $M$, which is how one shows random features or embeddings preserve magnitudes.
• Random projections. Projecting a fixed vector with a random matrix preserves its length up to a factor $1 \pm \varepsilon$. The fluctuation of $\|\Pi x\|^2$ is a quadratic form, and Hanson–Wright supplies the failure probability that feeds the Johnson–Lindenstrauss lemma.
• Covariance estimation. Controlling $x^\top(\hat\Sigma - \Sigma)x$, the error of a sample covariance in a fixed direction, is a quadratic form in the data.
• As a lemma. It is the usual way to show $x^\top A x$ is close to its mean for one $x$, then combined with the ε-net argument to make the statement hold uniformly over the whole sphere of directions.

References

• David L. Hanson, Farroll T. Wright. A bound on tail probabilities for quadratic forms in independent random variables. The Annals of Mathematical Statistics, 42(3):1079–1083, 1971. The original.
• Mark Rudelson, Roman Vershynin. Hanson-Wright inequality and sub-gaussian concentration. Electronic Communications in Probability, 18, no. 82, 1–9, 2013. The modern statement with the two norms and a clean proof by decoupling.
• Roman Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, 2018. Chapter 6 states and proves the inequality; the sub-exponential and Bernstein bounds it rests on are developed in Chapter 2.
• Stéphane Boucheron, Gábor Lugosi, Pascal Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013. Background on sub-gamma variables and the Bernstein bound behind the two regimes.