Maximum expected range under uncorrelated and dependent noise
For an integer , let be real-valued random variables such that and . Determine the sharp constant such that
under each prescribed dependence condition on the variables, including arbitrary dependence, independence, and pairwise nonnegative correlations. In particular, determine the optimal bound in the general dependence setting and establish whether the claimed sharp constants for independent, not necessarily identically distributed variables and for pairwise nonnegatively correlated variables are correct.
References
- Sharp Bounds on the Winner's Bias under Independent and Nonnegatively Correlated Noise — arXiv — Ivan Viakhirev, Kirill Borodin, Grach Mkrtchian
Source: arXiv
Progress summary
A new preprint claims exact answers in two important noise-dependence cases, but the broader problem remains open and the claims have not been independently verified.
The problem asks for the largest possible expected range of noisy observations under specified dependence assumptions. The general dependence problem is not settled.
Known results
- Papadatos (2014; published 2016) derived the best possible expected-range bound for arbitrary dependence when means and variances are specified, and characterized extremal distributions.
October 22, 2026 claimed progress
Ivan Viakhirev, Kirill Borodin, and Grach Mkrtchian claim sharp constants, extremizers, and related cross-validation bounds for independent and nonnegatively correlated noise. This is a claimed advance on two regimes, not a solution of the general problem.
Current status (as of October 2026): A preprint claims progress on two dependence regimes, while the general maximum-expected-range problem remains open and the new results are unverified.
Solutions 0
No solutions have been posted yet.