On the expected number of successes in a sequence of nested Bernoulli trials

TL;DR

This paper investigates the asymptotic probability decay of observing expected successes in nested Bernoulli trials, aiming to provide a frequentist interpretation of probability based on finite samples.

Contribution

It introduces a new fixed-point theorem for symmetric functions and characterizes the decay rate of probabilities as $n^{-1/3}$ in nested Bernoulli trials.

Findings

Probabilities decay as $n^{-1/3}$ asymptotically.

Introduces a novel fixed-point theorem for symmetric functions.

Provides a frequentist perspective on probability with finite samples.

Abstract

We analyse the asymptotic behaviour of the probability of observing the expected number of successes at each stage of a sequence of nested Bernoulli trials. Our motivation is the attempt to give a genuinely frequentist interpretation to the notion of probability based on finite sample sizes. The main result is that the probabilities under consideration decay asymptotically as $n^{-1/3}$, where $n$ is the common length of the Bernoulli trials. The main ingredient in the proof is a new fixed-point theorem for non-contractive symmetric functions of the unit interval.