The cosine integral against a centred Gaussian

The cosine integral against a centred Gaussian. For every $v \ge 0$,

Why this identity is useful. $\cos$ is a bounded $1$-Lipschitz function, so it is admissible both as a test function for weak convergence and for the elementary bound $|\cos a - \cos b| \le |a-b|$. The identity above says that integrating this one test function against $N(0,v)$ recovers $v$ injectively. That turns an estimate on laws into an estimate on variances: if two centred Gaussians are close when tested against $\cos$, their variances are close, provided they range in a bounded set. This is precisely the mechanism used to show that the truncated asymptotic variances in the Markov chain central limit theorem form a Cauchy sequence.

Proof. This is the real part of the characteristic function at $t = 1$. The characteristic function of $N(\mu, v)$ is $t \mapsto \exp(it\mu - vt^2/2)$; at $\mu = 0$, $t = 1$ it equals $e^{-v/2}$, a real number. On the other hand $\varphi(1) = \int e^{ix}\,dN(0,v)(x)$, and the integrand is bounded in modulus by $1$, hence integrable against a probability measure; taking real parts commutes with the Bochner integral for integrable functions, and $\operatorname{Re} e^{ix} = \cos x$. Comparing gives the claim.