Let ZZ be a Chi-squared random variable with kk degrees of freedom.

Pr[|𝔼Z−Z|≥ϵ𝔼Z]≤2e−kϵ2/8\mathrm{Pr}[|\mathbb{E}Z-Z| \geq \epsilon\mathbb{E}Z] \leq 2 e^{-k\epsilon^2/8}

See: concentration inequality

Reference:

  1. https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf
  2. https://en.wikipedia.org/wiki/Chi-squared_distribution