Functional bootstrap: a hardware constrained implementation of on-line bootstrap

We cope with the key step of bootstrap methods of generating a possibily infinite sequence of random data preserving properties of the distribution law, starting from a primary sample actually drawn from this distribution. We deal with two hardware resource constraints: i. absence of a long term memory, requiring an on line estimation of the bootstrap generator parameters, and ii. limited amount of mass memory, binding the number of statistics that can be collected at run-time. We use a probabilistic Random Access Memory (pRAM) neural network as a suitable hardware with the mentioned constraints, and we split the bootstrap sampling into the generation of many bernoullian variables. Each variable, since represents the random value of a single bit conditioned by the values assumed by others, identify its statistics with the content of the addresses the pRAM memory. On this hardware, on-line estimation has been obtained by a learning by gossip model which properly manages te run-time values of correlated estimating processes. An entropic rule has been user for decimating the conditional distributions to a number storable into the pRAM memory. Since the law of the bootstrap sample is now determined by the inner sructure of the trained hardware, we speak of the functional bootstrap. Phisical limitations, open technical problems, extensibility and effectiveness of the method are discussed and exhibited through numerical examples.