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 $\mathbb E_{Y_n}[f(X_n,Y_n)] = a(X_n)+o_{n,\mathbb P}(1)$ with $a(X_n) \to \alpha$ (a.s), 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 modification of the question which does ?