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| Formato: | Preprint |
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2026
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| Acceso en línea: | https://arxiv.org/abs/2601.19323 |
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| _version_ | 1866918308101488640 |
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| author | Fill, James Allen Janson, Svante |
| author_facet | Fill, James Allen Janson, Svante |
| contents | 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 $π$ is spherically symmetric and unimodal in the specific sense that $π({\bf x})=\hatπ(\|{\bf x}\|)$ for some nonincreasing function $\hatπ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19323 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive autocorrelation at unit lag for stationary random walk Metropolis-Hastings in ${\mathbb R}^d$ Fill, James Allen Janson, Svante Probability Statistics Theory 60J05 (Primary) 60J22, 60E15 (Secondary) 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 $π$ is spherically symmetric and unimodal in the specific sense that $π({\bf x})=\hatπ(\|{\bf x}\|)$ for some nonincreasing function $\hatπ$ 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. |
| title | Positive autocorrelation at unit lag for stationary random walk Metropolis-Hastings in ${\mathbb R}^d$ |
| topic | Probability Statistics Theory 60J05 (Primary) 60J22, 60E15 (Secondary) |
| url | https://arxiv.org/abs/2601.19323 |