Boundaries of the precision of restoring information lost after rounding the results from observations (Q682218)

Assume that \(X_1,X_2,\dots\) be a sequence of iid random quantities with unknown expectation \(\mu\), and \(\epsilon_1, \epsilon_2, \dots\) be a sequence distributed according one of the following distributions: normal, Laplace or Simpson with dispersion \(\sigma^{2}\). Consider the sequence of rounded data \((X_1+\epsilon_1)^*, (X_2+\epsilon_2)^*, \dots\), where the rounded value of \(x\) up to the closest integer is denoted by \(x^{*}\). Upper limits are given for \[ \Delta(\mu,\sigma)=\Big|\lim_{n\to\infty} \frac1n \sum_{i=1}^n (X_i+\epsilon_i)^*-\mu\Big| \] (where \(\lim\) means the almost certain limit). Analogous lower bounds are derived for \[ \Delta(\sigma)=\sup_\mu \Big|\lim_{n\to\infty} \frac1n \sum_{i=1}^n X_i^*-\mu\Big|. \]