Metrics and Norms on Spaces of Random Variables

Abstract

Let $\mathfrak{X}$ be the space of random variables defined on an abstract probability space $(\Omega, \mathcal{a}, P)$ where we consider any two elements of $\mathfrak{X}$ which are equal a.s. (almost surely) as the same. Frechet [2] exhibited a metric on $\mathfrak{X}$ (for example, $E\lbrack|X - Y|/(1 + |X - Y|)\rbrack)$ with the property that convergence in the metric is equivalent to convergence in probability, and he showed that for some probability spaces the same cannot be done for convergence a.s. Dugue [1] showed that it is not in general possible to define a norm on $\mathfrak{X}$ such that convergence in the norm is equivalent to convergence in probability. These results are contained in and completed by the following fact which was stated without proof by the author in [5] and which follows easily from the two theorems stated and proved in this note. There exists a metric (norm) on $\mathfrak{X}$ with convergence in the metric (norm) equivalent to convergence a.s. (in probability) if, and only if, $\Omega$ is the union of countable (finite) number of disjoint atoms. After these results were obtained it was found that the equivalence of parts (ii) and (iii) of Theorem 1 had been proved by Marczewski [4], p. 121. An atom of a probability space is a measurable set $A$ with $P(A) > 0$, such that any measurable subset has probability 0 or $P(A)$. It is easy to show that a random variable is a.s. constant on an atom. $f$ will always designate a real-valued function defined on $\mathfrak{X}$. Convergence in $f$ is said to be equivalent to convergence a.s. (in probability) if, for every sequence $\{X_n\}$ of elements from $\mathfrak{X}, f(X_n) \rightarrow 0$ if, and only if, $X_n \rightarrow 0$ a.s. (in probability). THEOREM 1. The following conditions on a probability space are equivalent. (i) There exists a function $f$, such that convergence in $f$ is equivalent to convergence a.s. (ii) For any sequence $\{X_n\}$ from $\mathfrak{X}$, if $X_n \rightarrow 0$ in probability, then $X_n \rightarrow 0$ a.s. (iii) $\Omega$ is a countable union of disjoint atoms. THEOREMS 2. The following conditions on a probability space are equivalent. (a) There exists a function $f$, such that convergence in $f$ is equivalent to convergence in probability and $f$ satisfies $|f(\alpha X)| = |\alpha|\cdot|f(X)|$ for any $X \varepsilon \mathfrak{X}$ and any real number $\alpha$. (b) $\Omega$ is a finite union of disjoint atoms.