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diff --git a/posts/2019-03-13-a-tail-of-two-densities.md b/posts/2019-03-13-a-tail-of-two-densities.md index 1d9cf75..a7a39cf 100644 --- a/posts/2019-03-13-a-tail-of-two-densities.md +++ b/posts/2019-03-13-a-tail-of-two-densities.md @@ -590,14 +590,14 @@ using $$\int_t^\infty e^{- {y^2 \over 2}} dy < \int_t^\infty {y \over t} e^{- {y^2 \over 2}} dy.$$ -The second is shown using Chernoff bound. For any random variable $\xi$, +The second is shown using [Chernoff bound](https://en.wikipedia.org/wiki/Chernoff_bound). For any random variable $\xi$, $$\mathbb P(\xi > t) < {\mathbb E \exp(\lambda \xi) \over \exp(\lambda t)} = \exp(\kappa_\xi(\lambda) - \lambda t), \qquad (6.7)$$ where $\kappa_\xi(\lambda) = \log \mathbb E \exp(\lambda \xi)$ is the cumulant of $\xi$. Since (6.7) holds for any $\lambda$, we can get the best bound by minimising $\kappa_\xi(\lambda) - \lambda t$ (a.k.a. the -Legendre transformation). When $\xi$ is standard normal, we get (6.5). +[Legendre transformation](https://en.wikipedia.org/wiki/Legendre_transformation)). When $\xi$ is standard normal, we get (6.5). $\square$ **Remark**. We will use the Chernoff bound extensively in the |