# The Smallest Probability Interval a Sequence Is Random for: A Study for Six Types of Randomness

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 12897)

## Abstract

There are many randomness notions. On the classical account, many of them are about whether a given infinite binary sequence is random for some given probability. If so, this probability turns out to be the same for all these notions, so comparing them amounts to finding out for which of them a given sequence is random. This changes completely when we consider randomness with respect to probability intervals, because here, a sequence is always random for at least one interval, so the question is not if, but rather for which intervals, a sequence is random. We show that for many randomness notions, every sequence has a smallest interval it is (almost) random for. We study such smallest intervals and use them to compare a number of randomness notions. We establish conditions under which such smallest intervals coincide, and provide examples where they do not.

### Keywords

• Probability intervals
• Martin-Löf randomness
• Computable randomness
• Schnorr randomness
• Church randomness

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1. 1.

$$\mathbb {N}$$ denotes the natural numbers and denotes the non-negative integers (A real $$x \in \mathbb {R}$$ is called negative, positive, non-negative and non-positive, respectively, if $$x<0$$, $$x>0$$, $$x\ge 0$$ and $$x \le 0$$).

2. 2.

Since $$\tau$$ is non-decreasing, it being unbounded is equivalent to $$\lim _{n \rightarrow \infty }\tau (n)=\infty$$.

## References

1. Ambos-Spies, K., Kucera, A.: Randomness in computability theory. Contemporary Math. 257, 1–14 (2000)

2. Bienvenu, L., Shafer, G., Shen, A.: On the history of martingales in the study of randomness. Electron. J. Hist. Probab. Stat. 5, 1–40 (2009)

3. De Cooman, G., De Bock, J.: Computable randomness is inherently imprecise. In: Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications. Proceedings of Machine Learning Research, vol. 62, pp. 133–144 (2017)

4. De Cooman, G., De Bock, J.: Randomness and imprecision: a discussion of recent results. In: Proceedings of the Twelfth International Symposium on Imprecise Probability: Theories and Applications. Proceedings of Machine Learning Research, vol. 147, pp. 110–121 (2021)

5. De Cooman, G., De Bock, J.: Randomness is inherently imprecise. Int. J. Approximate Reasoning (2021). https://www.sciencedirect.com/science/article/pii/S0888613X21000992

6. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010). https://doi.org/10.1007/978-0-387-68441-3

7. Persiau, F., De Bock, J., De Cooman, G.: A remarkable equivalence between non-stationary precise and stationary imprecise uncertainty models in computable randomness. In: Proceedings of the Twelfth International Symposium on Imprecise Probability: Theories and Applications. Proceedings of Machine Learning Research, vol. 147, pp. 244–253 (2021)

8. Persiau, F., De Bock, J., De Cooman, G.: The smallest probability interval a sequence is random for: a study for six types of randomness (2021). https://arxiv.org/abs/2107.07808, extended online version

9. Schnorr, C.P.: A unified approach to the definition of random sequences. Math. Syst. Theory 5, 246–258 (1971)

10. Wang, Y.: Randomness and Complexity. PhD thesis, Ruprecht Karl University of Heidelberg (1996)

## Acknowledgments

Floris Persiau’s research was supported by FWO (Research Foundation-Flanders), project number 11H5521N.

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### Corresponding authors

Correspondence to Jasper De Bock or Gert de Cooman .

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