Concentration results for non-standard Gaussian random vectors.

Question

Given a $c$-Lipschitz function $f(X):\mathbb{R}^d \rightarrow \mathbb{R}$, and given that $X \in \mathbb{R}^d$ is a Gausssian random vector centered at $\mathbb{w} \in \mathbb{R}^d$ (not at zero) having a co-variance matrix of $\sigma^2 \mathbb{I}$, what can we say about $ P( |f(X) - {\Bbb E} f(X)|\geq t)$? I know the results are standard for the case when every component of $X$ is distributed i.i.d as $N(0,1)$, but am looking for a more general result.

Thanks very much, MK

Answer 1

Since your covariance matrix is merely a multiple of the identity matrix, the result for standard normal should be easily adapted to your case by a scaling argument. I suspect this is a homework problem. But it seems too easy for such advanced topic.