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 Yn and 2 Y n+1 .