Research Report, Université du littoral côte d'Opale.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

- For online information, click here.

**Abstract:** Let Y_n be the number of attempts needed to get the nth success in a nonstationary sequence of independent Bernoulli trials and denote by α a fixed irrational number. We prove that, under mild conditions on the probabilities of success, the law of the fractional part of αY_n converges weakly to the uniform distribution on [0, 1) whenever α is irrational. We then compute upper bounds of the convergence rates depending on a measure of irrationality of α and on the probabilities of success. As an application, we discuss the mantissa of a Yn for positive integer a and the mantissa of the nth random Mersenne number generated by the Cramér model of pseudo-primes.

**Bibtex:**

```
@techreport{,
TITLE = {{Random walks on the circle and measure of irrationality}},
AUTHOR = {Mass{\'e}, Bruno},
URL = {https://hal.archives-ouvertes.fr/hal-03274061},
TYPE = {Research Report},
INSTITUTION = {{Universit{\'e} du littoral c{\^o}te d'Opale}},
YEAR = {2021},
MONTH = Dec,
PDF = {https://hal.archives-ouvertes.fr/hal-03274061v2/file/Random%20walks%20on%20the%20circle%20%28IJNT%29%20copie.pdf},
HAL_ID = {hal-03274061},
HAL_VERSION = {v2},
}
```

**Reference Type:** Technical Report

**Subject Area(s):** Probability Theory