| Summary: | We consider a Metropolis–Hastings method with proposal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mi>G</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>x</i> is the current state, and study its ergodicity properties. We show that suitable choices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> can change these ergodicity properties compared to the Random Walk Metropolis case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>Σ</mo><mo>)</mo></mrow></semantics></math></inline-formula>, either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis.
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