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I asked this question a couple of days ago on Math.SE but without any echo (no upvotes, although I offered a bounty). But because I did it for oversight from a reputable/professional source I now think meta.MO might be the more suitable place to ask. I hope you don't mind the cross-posting.


Assume ${\bf x},{\bf z} \in \mathbb{R}^n$ denote real-valued and bounded random variables with continuous probability density $p({\bf x},{\bf z})$ and $f({\bf x})$ and $g({\bf x})$ are real-valued bounded scalar functions. Furthermore, $P_k({\bf z})$ denote all monomials for ${\bf z}$ indexed by $k\in\mathbb{N}$ (so that the expectation values of $E[P_k]$ are the corresponding mixed moments of $\bf z$).

Then, does the following implication hold?

For all $k$: $ E\left[f({\bf x})P_k({\bf z})\right] = E\left[g({\bf x})P_k({\bf z})\right]~~~ \implies ~~~E\left.\left[f({\bf x})\right|{\bf z}\right] = E\left.\left[g({\bf x})\right|{\bf z}\right]$

Note: All moments and expectation values exist because the random variables and functions are bounded.


I tried the following. (I'm not so sure with the exchange of the limits and the uniform convergence.)

Because the premise holds for all polynomials of $\bf z$ I can represent for each $\epsilon$ the $\eta_\varepsilon({\bf z})$ (Gaussian bump representation) in the Dirac-Delta function in $\mathbb{R}^n$

$\delta({\bf z})=\lim_{\varepsilon\to 0^+} \eta_\varepsilon({\bf z})$

as the powerseries of the exponential function in the Gaussian bump representation with coefficients $c_k$. This powerseries converges uniformly to the exponential function and, thus, the arguments of the expectation values converge uniformly to $f({\bf x})\eta_\varepsilon({\bf z})$ and $g({\bf x})\eta_\varepsilon({\bf z})$ (for arbitrary bounded $f$ and $g$). Then, I can exchange the integral of the expectation with the infinite sum and obtain

$$ \sum_k c_k E\left[f({\bf x})P_k({\bf z})\right] = \sum_k c_k E\left[g({\bf x})P_k({\bf z})\right] \\ E\left[f({\bf x})\sum_k c_k P_k({\bf z})\right] = E\left[g({\bf x})\sum_k c_kP_k({\bf z})\right]\\ E\left[f({\bf x})\eta_\varepsilon({\bf z} -{\bf z}_0)\right] = E\left[g({\bf x})\eta_\varepsilon({\bf z} -{\bf z}_0)\right]\\ \lim_{\varepsilon\to 0^+}E\left[f({\bf x})\eta_\varepsilon({\bf z} -{\bf z}_0)\right] = \lim_{\varepsilon\to 0^+}E\left[g({\bf x})\eta_\varepsilon({\bf z} -{\bf z}_0)\right]\\ E\left[f({\bf x})\delta({\bf z} -{\bf z}_0)\right] = E\left[g({\bf x})\delta({\bf z} -{\bf z}_0)\right]\\ E\left[f({\bf x})|{\bf z}_0\right] = E\left[g({\bf x})|{\bf z}_0\right]$$

Is this correct?

In general, I am interested if the statement holds without the assumption of a continuous probability density $p({\bf x},{\bf z})$ for arbitrary probability distributions, but I don't see any reason why it should. Any ideas?

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  • $\begingroup$ A useful tool for proving results of this kind is the multiplicative system theorem. $\endgroup$ May 20, 2020 at 23:57

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Your desired conclusion does hold without the assumption of a probability density of $\bf x$ and $\bf z$, provided that the functions $f$ and $g$ are assumed to be Borel measurable.

Indeed, let $X:=f(\bf x)-g(\bf x)$ and $Z:=\bf z$, so that $X$ and $Z$ are bounded random variables with values in $\mathbb R$ and $\mathbb R^n$, respectively, such that $$EXP(Z)=0$$ for all polynomials $P$ on $\mathbb R^n$. Since the set of all such polynomials is dense in $L^1(K)$ for any compact $K\subset\mathbb R^n$, whereas $X$ and $Z$ are bounded, we conclude that $EXh(Z)=0$ for any $h\in L^1(K)$. So, $EX\,1(Z\in B)=0$ for any Borel set $B\subseteq\mathbb R^n$, which means $E(X|Z)=0$, as desired.

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  • $\begingroup$ Great, thanks. Yes, my $f$ and $g$ are continuous functions and thus Borel measurable. Do I get it right that your proof holds for any probability distribution with compact support (+ the constraints for $f$ and $g$)? $\endgroup$
    – Its_me
    May 21, 2020 at 11:45
  • $\begingroup$ I've seen that you also have a math.SE account. If you are interested in a math.SE 100 points bounty you could repost your answer there and I'll assign it to you. $\endgroup$
    – Its_me
    May 21, 2020 at 11:47
  • $\begingroup$ Could you please help me with your statement "It is unclear how you define the expressions $E\left[f({\bf x})\delta({\bf z} -{\bf z}_0)\right]$ and $E\left[f({\bf x})|{\bf z}_0\right]$". $E\left[f({\bf x})|{\bf z}_0\right]$ should exist since $E[f({\bf x})]$ exists (by Radon–Nikodym theorem) and is almost surely unique. And $E[f({\bf x})\delta({\bf z} -{\bf z}_0)] := \lim_{\varepsilon\to 0^+}\int f({\bf x})\eta_\varepsilon({\bf z} -{\bf z}_0) p({\bf x},{\bf z})d{\bf x}d{\bf z} = \int f({\bf x})p({\bf x},{\bf z}_0)d{\bf x}=E[f({\bf x})|{\bf z}_0]$ for density $p({\bf x},{\bf z})$, isn't it? $\endgroup$
    – Its_me
    May 21, 2020 at 12:13
  • $\begingroup$ @Its_me : Concerning your first comment: Yes, in $\mathbb R^n$ compact support and bounded support mean the same, because the support set is closed, and a subset of $\mathbb R^n$ is compact if and only if it is bounded and closed. Concerning your second comment: I have posted this answer on math SE. Thank you for this suggestion. Concerning your third comment: Sorry, I did not notice that you are assuming the existence of a joint pdf of $\bf x$ and $\bf z$. I have edited my answer accordingly. $\endgroup$ May 21, 2020 at 13:12
  • $\begingroup$ I went again through the proof and afaik it applies also to Borel measurable functions that depend on both r.v., i.e. $f({\bf x},{\bf z})$ and $g({\bf x},{\bf z})$, doesn't it? $\endgroup$
    – Its_me
    May 29, 2020 at 12:31

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