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An infinite sequence x  1  x  2  …  of letters from some alphabet {  0  ,  1  ,  …  ,  b  −  1  }  , b  ≥  2  , is called k-distributed (  k  ≥  1  ) if any k-letter block of successive digits appears with the frequency b  −  k  in the long run. The sequence is called normal (or ∞-distributed) if it is k-distributed for any k  ≥  1  . We describe two classes of low-entropy processes that with probability 1 generate either k-distributed sequences or ∞-distributed sequences. Then, we show how those processes can be used for building random number generators whose outputs are either k-distributed or ∞-distributed. Thus, these generators have statistical properties that are mathematically proven.

mathematics

Low-Entropy Stochastic Processes for Generating k-Distributed and Normal Sequences, and the Relationship of These Processes with Random Number Generators †