### Massé, B (2021)

#### Random subsequences of αn with asymptotically
independent successive terms

Research Report, Université du Littoral - Côte d'Opale.

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

**Abstract:** Fix an irrational number α and let Y_n be the number of attempts needed to get the nth success in a non-stationary sequence of independent Bernoulli trials. It is known that the law of the fractional part of αY_n converges weakly to the uniform distribution on [0, 1) , as n → +∞, when the probabilities of success decrease to 0 and sum to +∞. We provide sufficient conditions on the probabilities of success ensuring that the fractional parts of αY n and αY n+1 are asymptotically independent. We extend our results to any number of successive terms, compute upper bounds of the convergence rates depending on a measure of irrationality of α and on the probabilities of success and apply our results to discuss the mantissae of 2 Y_n and 2 Y_n+1 .

**Bibtex:**

```
@techreport{,
TITLE = {{Random subsequences of \{$\alpha$n\} with asymptotically independent successive terms}},
AUTHOR = {Mass{\'e}, Bruno},
URL = {https://hal.archives-ouvertes.fr/hal-03502439},
TYPE = {Research Report},
INSTITUTION = {{Universit{\'e} du Littoral - C{\^o}te d'Opale}},
YEAR = {2021},
MONTH = Dec,
PDF = {https://hal.archives-ouvertes.fr/hal-03502439/file/Asymptotic%20independence%202%20copie.pdf},
HAL_ID = {hal-03502439},
HAL_VERSION = {v1},
}
```

**Reference Type:** Technical Report

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