The Annals of Mathematical Statistics

Conditional Expectations of Random Variables Without Expectations

R. E. Strauch

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Abstract

Let $(\Omega, \mathscr{F}, P)$ be a probability space, and let $X$ be a random variable defined on $(\Omega, \mathscr{F}, P)$. If $\mathscr{A}$ is a sub $\sigma$-field of $\mathscr{F}$, then $E(X \mid \mathscr{A})$ is the a.s. unique $\mathscr{A}$ measurable function such that, for all $A \varepsilon \mathscr{A}$, \begin{equation*}\tag{1}\int_A X dP = \int_A E(X \mid \mathscr{A}) dP,\end{equation*} provided $EX$ is defined. ([2], p. 341). If $EX$ is not defined, that is, if $EX^+ = EX^- = \infty$, we may then define $E(X \mid \mathscr{A}) = E(X^+ \mid \mathscr{A}) - E(X^- \mid \mathscr{A})$, provided the difference is defined almost surely ([2], p. 342). We show that this is the only reasonable definition of $E(X \mid \mathscr{A})$ (Lemma 2), and exhibit several apparent pathologies, akin to the fact that a conditionally convergent series of real numbers may be re-ordered to give any sum. If $X$ is any random variable with a continuous distribution such that $EX$ is not defined, then we can find an $\mathscr{A} \subset \mathscr{F}$ such that $E(X \mid \mathscr{A}) = 0$ a.s. (Theorem 1), and if $Y$ is any random variable independent of $X$, we can find an $\mathscr{A} \subset \mathscr{F}$ such that $E(X \mid \mathscr{A}) = Y$ a.s. (Theorem 2). In fact, if $X_1, X_2, \cdots$ is a sequence of independent random variables such that for $n \geqq 2, EX_n$ is not defined and $X_n$ has a continuous distribution, we can find a sequence of $\sigma$-fields $\mathscr{A}_1 \subset \mathscr{A}_2 \subset \cdots \subset \mathscr{F}$ such that $X_1, \cdots, X_n$ are $\mathscr{A}_n$ measurable and $E(X_{n + 1} \mid \mathscr{A}_n) = X_n$ a.s. (Theorem 3). The sequence $\{X_n, \mathscr{F}_n, n = 1, 2, \cdots\}$ is not a martingale however, since for $m > n + 1, E(X_m \mid \mathscr{F}_n)$ is not defined. We remark that the standard theorem on iterated conditional expectations, ([2], p. 350) which says that if $\mathscr{A} \subset \mathscr{B} \subset \mathscr{F}$ then $E(X \mid \mathscr{A}) = E(E(X \mid \mathscr{B}) \mid \mathscr{A})$ a.s. is valid only if $E(X \mid \mathscr{A})$ is defined a.s.

Article information

Source
Ann. Math. Statist., Volume 36, Number 5 (1965), 1556-1559.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699915

Digital Object Identifier
doi:10.1214/aoms/1177699915

Mathematical Reviews number (MathSciNet)
MR181002

Zentralblatt MATH identifier
0138.10604

JSTOR
links.jstor.org

Citation

Strauch, R. E. Conditional Expectations of Random Variables Without Expectations. Ann. Math. Statist. 36 (1965), no. 5, 1556--1559. doi:10.1214/aoms/1177699915. https://projecteuclid.org/euclid.aoms/1177699915


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