Summary
It seems empirically that the first digits of random numbers do not occur with equal frequency, but that the earlier digits appear more often than the latters. This peculiality was at first noticed by F. Benford, hence this phenomenon is called Benford's law.
In this note, we fix the set of all positive integers as a model population and we sample random integers from this population according to a certain sampling procedure. For polynomial sampling procedures, we prove that random sampled integers do not necessarily obey Benford's law but their Banach limit does. We also prove Benford's law for geometrical sampling procedures and for linear recurrence sampling procedures.
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Nagasaka, K. On Benford's law. Ann Inst Stat Math 36, 337–352 (1984). https://doi.org/10.1007/BF02481974
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DOI: https://doi.org/10.1007/BF02481974