Stochastic Search in a Banach Space
P. Warwick Millar · Birkhäuser Boston eBooks · 1992
This paper establishes a probabilistic result which has implications for the numerical implementation of certain non-parametric statistical procedures. To describe the probabilistic result, let λn, n ≥ 1 be an increasing sequence of integers, totally arbitrary except for the condition that λn ↑ ∞. Let Y 1, Y 2,… be i.i.d. random variables with values in an infinite dimensional Banach space B, and whose common distribution μ has as its support the unit ball of B. Then no matter what ∈ > 0 is selected, and no matter which θ 0 in the unit ball of B is chosen, and no matter what the rate at which λn ↑ ∞, one clearly has $$\lim_{n\rightarrow\infty}P\left \{|Y_{i}-\theta_{0}|\leq\, e\, for some\, i\leq\lambda_{n}\right \}=1$$