Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function.

>**Question 1.** If we establish that $\mathbb E_{Y_n}[f(X_n,Y_n)] = \alpha+o_{n,\mathbb P}(1)$ (for some $\alpha \in \mathbb R$), and $var_{Y_n}(f(X_n,Y_n)) = o_{n,\mathbb P}(1)$, can one conclude that $f(X_n,Y_n) = \alpha + o_{n,\mathbb P}(1)$ without further assumptions ?

Naively, I'd say **Yes**, by Chebyshev's inequality. But I worry that something strange might be going on in general, to require a bit more care.

>**Question 2.** In case Question 1 does not answer in the affirmative, is there a "slight" modification thereof which does ?