Positive autocorrelation at unit lag for stationary random walk Metropolis-Hastings in ${\mathbb R}^d$

It is often asserted in the literature that one should expect positive autocorrelation for random walk Metropolis-Hastings (RWMH), especially if the typical proposal step-size is small relative to the variability in the target density. In this paper, we consider a stationary RWMH chain ${\bf X}$ taking values in $d$-dimensional Euclidean space and (subject only to the existence of densities with respect to Lebesgue measure) with general target distribution having finite second moment and general proposal random walk step-distribution. We prove, for any nonzero vector ${\bf c}$, strict positivity of the autocorrelation function at unit lag for the stochastic process $\langle{\bf c},{\bf X}\rangle$, that is, $${\operatorname{Corr}}(\langle{\bf c},{\bf X}_0\rangle,\langle{\bf c},{\bf X}_1\rangle)>0,$$ and we establish the same result, but with weak inequality (which can in some cases be equality) when the state space for ${\bf X}$ is changed to the integer grid ${\mathbb Z}^d$. Further, for ${\bf c}\neq{\bf 0}$ we establish the sharp lower bound $${\operatorname{Corr}}(\langle{\bf c},{\bf X}_0\rangle,\langle{\bf c},{\bf X}_1\rangle)>\tfrac19$$ on autocorrelation when we assume both that (i) the target density $\pi$ is spherically symmetric and unimodal in the specific sense that $\pi({\bf x})=\hat{\pi}(\|{\bf x}\|)$ for some nonincreasing function $\hat{\pi}$ on $[0,\infty)$ and that (ii) the proposal step-density is symmetric about ${\bf 0}$. We study the autocorrelation indirectly, by considering the incremental variance function (or incremental second-moment function) at unit lag. The same approach allows us also for $r\in[2,\infty)$ to upper-bound the incremental $r$th-absolute-moment function at unit lag. We give also closely related inequalities for the total variation distance between two distributions on ${\mathbb R}^d$ differing only by a location shift.