Functional continuous mapping theorem

Question

Consider a sequences of stochastic processes $(f_n)$ on $[0,1]$. If we know that, $\forall u \in [0,1], \, f_n(u)=o_p(1)$ (that is: pointwise convergence to zero in probability), since the integral is a continuous operator, can we conclude by continuous mapping theorem that $\int_0^1f_n(u) du \overset{p}{\to} 0$? My impression is: no, we need a stronger hypothesis on $(f_n)$, such as $\Vert f_n \Vert_\infty \overset{p}{\to}0$. Am I wrong?

Answer 1

It is indeed not sufficient: consider the case $f_n(u)=n\mathbf 1\left\{0\lt u\lt 1/n\right\}$ for $n\geqslant 1$. We have for each $u$ that $f_n(u)=0$ for $n$ large enough but $\int_0^1f_n(u)\mathrm du=1$ for each $n$.

What could help in this context is a use of uniform integrability: if we assume that with probability one, $\lim_{R\to +\infty}\sup_{n\geqslant 1}\int_0^1 \left|f_n(u) \right| \mathbf 1 \left\{\left|f_n(u) \right| \gt R\right\}\mathrm du=0$. In this case, the result can be used even if the process $f_n$ is not bounded with respect to $u$.