# Inverse marginal property of a collection of $\sigma$-algebras

In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" I introduced the Inverse marginal property (IMP) for a collection of $$\sigma$$-algebras.

Let $$(\Omega,\mathcal{F},\mu)$$ be a probability space and $$\mathcal{F}_1,...,\mathcal{F}_n$$ be sub-$$\sigma$$-algebras of $$\mathcal{F}$$. We will say that the collection $$\mathcal{F}_1,...,\mathcal{F}_n$$ possesses the inverse marginal property (IMP) if for arbitrary random variables $$\xi_1,...,\xi_n$$ such that

(1) $$\xi_k$$ is $$\mathcal{F}_k$$-measurable, $$k=1,2,...,n$$;

(2) $$E|\xi_k|^2<\infty$$, $$k=1,2,...,n$$;

(3) $$E\xi_1=E\xi_2=...=E\xi_n$$,

there exists a random variable $$\xi$$ such that $$E|\xi|^2<\infty$$ and $$E(\xi|\mathcal{F}_k)=\xi_k$$ for all $$k=1,2,...,n$$.

The simplest example of a collection of sub-$$\sigma$$-algebras which possesses the IMP is a system of pairwise independent sub-$$\sigma$$-algebras. In this case a needed random variable $$\xi$$ can be defined by $$\xi:=\xi_1+...+\xi_n-(n-1)a$$, where $$a:=E\xi_1=E\xi_2=...=E\xi_n$$.

Question: is the IMP a new notion or it is well-known? Have you seen this property or something similar in the literature?

I will be very grateful for any comments on the IMP.

• If one restrict to $\xi$ such that $\mathbb{E}(\xi)=0$. Is your IMP equivalent to $$L^2(\Omega,\mathcal{F}_1,\mu)\oplus L^2(\Omega,\mathcal{F}_2,\mu)\oplus\cdots \oplus L^2(\Omega,\mathcal{F}_n,\mu)$$? Apr 30, 2020 at 15:38
• Unfortunately, I do not understand your question. Please specify the question. Apr 30, 2020 at 18:48
• Just that the vector space generated by the variable $\mathcal{F}_i$ measurable are in direct sum : $\forall \xi_1,\cdots,\xi_n\in L^2(\Omega,\mathcal{F_1},\mu)\times...\times L^2(\Omega,\mathcal{F_n},\mu)$. $\xi_1+\cdots +\xi_n=0\Rightarrow \xi_1=0,\cdots,\xi_n=0$. Apr 30, 2020 at 20:12
• No, IMP is not equivalent to the linear independence of the marginal subspaces $L^2_0(\mathcal{F}_1),...,L^2_0(\mathcal{F}_n)$. One can show that a collection of $\sigma$-algebras $\mathcal{F}_1,...,\mathcal{F}_n$ possesses the IMP if and only if the marginal subspaces $L^2_0(\mathcal{F}_1),...,L^2_0(\mathcal{F}_n)$ are linearly independent and their sum is closed in $L^2(\mathcal{F})$. For details see my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" (available on ArXiv). Apr 30, 2020 at 21:15
• Please choose between "article" and "paper" once and for all, and stop making these needless changes. May 3, 2020 at 4:34

I've never heard the term 'inverse marginal property', but the notion is somewhat familiar. Take any square integrable martingale $$\xi_i$$, $$i=1,2,...$$, and let $$\{\mathcal{F}_n \}$$ be it's natural filtration. Then I suspect for any $$N < \infty$$, the initial sequence $$\{ \mathcal{F}_n, n \leq N \}$$ has this property.
The difference is that $$\xi_n$$ are not arbitrary $$\mathcal{F}_n$$ measurable random variables - they're rather special. I expect that given our filtration, it is easy to construct a sequence $$\{ \zeta_n \}$$ adapted to $$\mathcal{F}_n$$ that break your IMP.
In fact, modifying your example slightly, $$\xi_{k} = a + \sum_{i\leq k}(\xi_i - a)$$, $$k = 1, \cdots, n$$ is a martingale w.r.t $$\{ \mathcal{F_k} \}$$. It is not an arbitrary sequence.
Could you construct a sequence of $$\sigma$$-algebras that satisfy IMP for any adapted sequence of random variables?
• In the definition of the IMP the needed $\xi$ must exist for arbitrary random variables $\xi_1,...,\xi_n$ that satisfy (1), (2), (3) in the Question. Apr 27, 2020 at 18:57
• If $\mathcal{F_1}\subset\mathcal{F}_2\subset...\subset\mathcal{F}_n$, then the collection $\mathcal{F}_1,...,\mathcal{F}_n$ does not possess the IMP (if the probability space is not trivial). In fact, if $\xi_1,...,\xi_n$ are random variables that satisfy conditions (1),(2),(3) in the Question, then the needed $\xi$ exists if and only if $\xi_1,...,\xi_n$ is a martingale with respect to $\{\mathcal{F}_1,...,\mathcal{F}_n\}$. If this is the case, then one can take $\xi=\xi_n$. Apr 27, 2020 at 19:02
• For examples of collections of $\sigma$-algebras that possess the IMP see Section 5 of my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" (it is available on ArXiv). Apr 27, 2020 at 19:06
• If $\mathcal{F}_1\subset...\subset\mathcal{F}_n$, then the collection $\mathcal{F}_1,...,\mathcal{F}_n$ does not possess the IMP (if the $\sigma$-algebras are not trivial). Apr 27, 2020 at 19:22