Testing independent sparse heterogeneous mixtures

We study the sequential discernibility between the independent fair coin tossing μ0 and the sparse heterogeneous mixtures HM(α,γ)≡(1-εn)·Bernoulli(1/2)+εn·Bernoulli((1+θn)/2) with εn∼n-γ and θn∼n-α. Extending the result in Lim (2003), we show that HM(α,γ) and μ0 are sequentially discernible when 0<α+γ⩽0.5, but are not so when 0.5<γ+α<1. It will be shown that each of the three different discernibility in [Lim, 2003. Testing stochastic processes: stationarity, independence, and ergodicity. Technical Report], (1) discernibility with an entire sample path, (2) uniformly discernibility with an entire sample path, and (3) sequentially discernibility, is equivalent to each other under this setting, and furthermore, it follows that sequential decision procedures are equivalent to the decisions based on an entire sequence. Finally, we show that the coin tossing with finitely many trials of biased coins is not sequentially discernible from that with infinitely many trials.