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Nov 22, 2023 at 20:30 vote accept Lau
Nov 22, 2023 at 15:45 comment added Stefaan Vaes I updated my answer and included two examples to show that this factor 2 is unavoidable and actually optimal.
Nov 22, 2023 at 15:45 history edited Stefaan Vaes CC BY-SA 4.0
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Nov 22, 2023 at 10:39 comment added Lau Thanks for the very nice answer Stefaan! And you didn't even use ergodicity. I'll leave the question open in hope of a proof/counterxample for the general case.
Nov 21, 2023 at 7:38 history answered Stefaan Vaes CC BY-SA 4.0