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authorYuchen Pei <me@ypei.me>2019-03-20 14:05:04 +0100
committerYuchen Pei <me@ypei.me>2019-03-20 14:05:04 +0100
commit31f1546d123eb571661f0e7373b7cae0d6b26dad (patch)
tree602bdfa551fc49d832e2e9be88ecb85fe880f569 /posts
parentbe929112fe305b5a9949d000cc29d03bccfb9db9 (diff)
minor
Diffstat (limited to 'posts')
-rw-r--r--posts/2019-03-13-a-tail-of-two-densities.md4
1 files changed, 2 insertions, 2 deletions
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