Counterexample Let X be a random variable taking on the values 1, 0, 1, 2, with probabilities P X 1 P X x (1 ) 2 x , x 0, 1, Then P P, 0 1 is boundedly complete but not complete Girschick et al (1946) ...

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Counterexample. Let X be a random variable taking on the values 1, 0, 1, 2, with

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Counterexample. Let X be a random variable taking on the values

−1, 0, 1, 2, … with probabilities Pθ{X = −1} = θ; Pθ{X = x} = (1 − θ)

θx , x = 0, 1,....

Then P = {Pθ, 0 < θ < 1} is boundedly complete but not complete. [Girschick et al. (1946)]

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Counterexample. Let X be a random variable taking on the values 1, 0, 1, 2, with

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