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Remarks on the space of monotonic functions. (English) Zbl 0564.60038

Let \(I_ m=\{k/m|\) \(k=0,...,m-1\}\) and \(F_{mn}\) be the set of nondecreasing functions \(f:I_ m\to I_ n\), and \(F_{m\infty}\) be the set of nondecreasing functions \(f:I_ m\to [0,1]\). In each \(F_{mn}\) we have the natural counting or m-dimensional Lebesgue probability measures. Let P(m,n,\(\epsilon)\) be the probability that some \(f\in F_{mn}\) satisfies \(| f(x)-x| <\epsilon\) for all \(x\in I_ m\). It is proved that \[ \lim_{m\to \infty}P(m,\infty,\epsilon)=1,\quad \lim_{m\to \infty}P(m,n,\epsilon)=1,\quad for\quad n\geq 2/\epsilon, \]
\[ \lim_{m,n\to \infty}P(m,n,\epsilon)=1,\quad \lim_{n\to \infty}P(m,n,\epsilon)=P(m,\infty,\epsilon). \] Remarks are made concerning the relationship of those facts, a theorem of Banach and Laplace’s principle of insufficient reason.

MSC:

60F99 Limit theorems in probability theory
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