Tensorization of a Poincaré inequality
= Tensorization of a Poincaré inequality
{c}
If each probability measure $\mu_i$ has Poincaré constant $C_i$, then the product measure $\bigotimes_i\mu_i$ has Poincaré constant at most $\max_iC_i$. This follows by iterating the <law of total variance> and applying <Jensen inequality> to derivatives of conditional expectations.