INDEPENDENT RANDOM VARIABLES ON THE UNIT INTERVAL

James Foran and Lee Hart

University of Missouri–Kansas City

Let the probability of a subset of [0, 1] be given by its Lebesgue measure, i.e., the uniform distribution. In this paper we relate independent random variables, which are continuous functions, to space filling curves. In modern terminology, a random variable is a real valued measurable function defined on a probability space. A collection V of random variables is said to be an independent collection if, for any natural number n and for any collection of functions {f_{1}, f_{2}, …, f_{n}} ⊂ V and Borel subsets (equivalently, open intervals) A_{1}, A_{2}, …, A_{n} of ℝ, we have

(1)  Pr (⋂_{i=1}^{n}{x | f_{i}(x) ∈ A_{i}}) = ∏_{i=1}^{n} Pr ({x | f_{i}(x) ∈ A_{i}}) .

We are interested in considering random variables defined on the probability space consisting of the unit interval with the probability of a set given by its Lebesgue measure. A classical example of a collection of independent random variables defined on [0, 1] is that of the Rademacher functions {f_{n}(x)}_{n=1}^{∞} where

f_{n}(x) = 1, if x ∈ [m/2^{n}, (m + 1)/2^{n})

with m even and

f_{n}(x) = −1, if x ∈ [m/2^{n}, (m + 1)/2^{n})

with m odd.

Considerable work has been done studying general measurable functions which are independent on [0, 1] and on the intervals [0, ∞) and ℝ = (−∞, ∞).

Many references to early work on this subject as well as the papers themselves, may be found in [4]. The possibility that such functions be continuous began with the observation that the coordinate functions of the Peano curve, which maps [0, 1] continuously into the unit square, are independent. (See [1] or [2] for recently discovered properties and a lucid description of the Peano curve.) Sierpinski [3] showed that if x(t) and y(t) are the coordinate functions of the Peano curve, then f_{n}(t) = x(y^{n}(t)), n = 0, 1, 2, … where y^{0}(t) = t, y^{n}(t) = y(y^{n−1}(t)) are independent and map [0, 1] onto [0, 1]^{ω}.

Most commonplace non–constant, continuous functions are not independent. In fact, if f_{1} and f_{2} are defined on [0, 1] and if there are intervals [a, b] in the range of f_{1} and [c, d] in the range of f_{2} so that f_{1}^{−1}([a, b]) ∩ f_{2}^{−1}([c, d]) = ∅, then f_{1} and f_{2} are not independent. This is because

m({x | f_{1}(x) ∈ [a, b]}) · m({x | f_{2}(x) ∈ [c, d]}) ≠ 0

and

m({x | f_{1}(x) ∈ [a, b] and f_{2}(x) ∈ [c, d]}) = 0.

This suggests that, in order that two non–constant continuous functions be independent on [0, 1], the functions must be somewhat unusual. This is borne out by the theorem below.

\underline{Theorem 1}. If f_{1} and f_{2} are continuous, non–constant independent functions defined on [0, 1] and if the range of f_{1} is [a, b] and the range of f_{2} is [c, d], then F (x) = (f_{1}(x), f_{2}(x)) is a continuous map from [0, 1] onto [a, b] × [c, d]; that is, F (x) is a “space–filling curve”.

\underline{Proof}. Suppose f_{1} and f_{2} are given as in the statement of the theorem and that (x_{0}, y_{0}) is any point in [a, b] × [c, d]. Let (a′, b′) and (c′, d′) be any two open intervals with x_{0} ∈ (a′, b′) and y_{0} ∈ (c′, d′). Since f_{1} and f_{2} are continuous, f_{1}^{−1}((a′, b′)) and f_{2}^{−1}((c′, d′)) are non–empty open sets and, hence, are of positive Lebesgue measure. Since f_{1} and f_{2} are independent,

\begin{matrix}m(\{x | f_{1}(x) ∈ (a′, b′) and f_{2}(x) ∈ (c′, d′)\}) \\ = m(\{x | f_{1}(x) ∈ (a′, b′)\}) · m(\{x | f_{2}(x) ∈ (c′, d′)\}) > 0. \end{matrix}

Since this is true, there is a point t such that F (t) = (f_{1}(t), f_{2}(t)) belongs to (a′, b′)×(c′, d′). Since (a′, b′) and (c′, d′) are arbitrary intervals satisfying x_{0} ∈ (a′, b′) and y_{0} ∈ (c′, d′) and since F (x) is continuous, it follows that there is t_{0} ∈ [0, 1] such that F (t_{0}) = (x_{0}, y_{0}). Then, (x_{0}, y_{0}) being an arbitrary point in [a, b] × [c, d], it follows that the range of F is [a, b] × [c, d] and thus, F is a “space–filling curve”.

Clearly, the same result holds true if Lebesgue measure is replaced by any non–atomic measure whose support is an interval [u, v]. Also it is clear that if {f_{n}} are finite or infinite sequence of independent random variables which are continuous on [a, b], then

F (x) = (f_{1}(x), f_{2}(x), …)

is a “space filling curve,” that is a continuous map from [a, b] to [u, v]^{n} or [u, v]^{ω}.

\underline{References}

1. G. Ene, “Peano Curves and Continuous Functions Whose Level Sets are Perfect,” Real Analysis Exchange, 3 (1987–8), 156–162.

2. J. Maly, “The Peano Curve and the Density Topology,” Real Analysis Exchange, 5 (1979–80), 326–330.

3. W. Sierpinski, “Remarque sur la Courbe Peannienne,” Wiad Matematyczne, 42 (1936), 1.

4. H. Steinhaus, Selected Papers, Polish Scientific Publishers, Warsaw, 1985.