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Comment : I think you could give more precision in the formulation of your question because you already have most of the answer.

In particular:

1-Is the generalization of Jensen equation you give valid on Hilbert Space ? (seems to be true, see http://www.emis.de/journals/JIPAM/images/075_02_JIPAM/075_02_www.pdf)

2-What assumption do you make on the random variables? do they have the same distribution ? are their distribution "equivakent" (i.e mutually absolutly continuous)

Answer: If (the answer to my question 1 is yes and if) $X$ and $Y$ have the same distribution $P$, because $L_2(P)$ is a hilbert space then you are done. Here, approximate linearity will mean that there exists a linear form $L$ such that $E[(f(X)-L(X))^2]E_P[(f(X)-L(X))^2]<\epsilon$.

If $X$ and $Y$ have different distribution (say $P_X$ and $P_Y$) I see an easy case: when $E_{P_Y}[( dP_X/dP_Y )^2] ) < c$ ($\chi^2$ divergence between distributions bounded). Indeed, in this case you can work in $L_2(P_Y)$ using cauchy swartz inequality. Otherwise the question needs to be clarifyed (i.e what do you mean by "approximatly linear"? : in what banach space do you choose to work ? ).

Hope this helps !

1

I think you could give more precision in the formulation of your question because you already have most of the answer.

In particular:

1-Is the generalization of Jensen equation you give valid on Hilbert Space ? (seems to be true, see http://www.emis.de/journals/JIPAM/images/075_02_JIPAM/075_02_www.pdf)

2-What assumption do you make on the random variables? do they have the same distribution ? are their distribution "equivakent" (i.e mutually absolutly continuous)

If (the answer to my question 1 is yes and if) $X$ and $Y$ have the same distribution $P$, because $L_2(P)$ is a hilbert space then you are done. Here, approximate linearity will mean that there exists a linear form $L$ such that $E[(f(X)-L(X))^2]<\epsilon$.

If $X$ and $Y$ have different distribution (say $P_X$ and $P_Y$) I see an easy case: when $E_{P_Y}[( dP_X/dP_Y )^2] ) < c$ ($\chi^2$ divergence between distributions bounded). Indeed, in this case you can work in $L_2(P_Y)$ using cauchy swartz inequality. Otherwise the question needs to be clarifyed (i.e what do you mean by "approximatly linear"? : in what banach space do you choose to work ? ).

Hope this helps !