Gaussian tail bound

For $X \sim \mathcal{N}(\mu,\sigma^2)$: $$\mathrm{Pr}[|X-\mathbb{E}X|\geq k \cdot \sigma] \leq 2e^{-k^2/2}$$

see Gaussian concentration

compare Chebyshev’s Inequality; Gaussian random variables concentrate much tighter around their expectation than variance alone (i.e. Chebyshevs’s inequality) predicts.

References:

1. https://www.math.uci.edu/~rvershyn/papers/concentration-random-tensors.pdf