Let $X$ be a random vector with probability distribution $P_X$ and support $\Pi \subset \mathbb{R}^d$. Suppose $\{f_n\}$ is a sequence of measurable random functions with domain $\Pi$ and codomain $\mathbb{R}^m$. Assume that the following terms are well defined, and the limits are as $n \to \infty$.
- Does $\mathbb{E}_X\left[\left\lVert f_n(X) \right\rVert^2\right] \xrightarrow{a.s.} 0 \implies f_n(X) \xrightarrow{p} 0$ for $P_X$-a.e. X?
- Does $\mathbb{E}_X\left[\left\lVert f_n(X) \right\rVert^2\right] \xrightarrow{p} 0 \implies f_n(X) \xrightarrow{p} 0$ for $P_X$-a.e. X?
My sense is that the first result is true because for the expectation of a positive random variable to be converging to zero with probability one, we require that the random variable itself converge to zero almost everywhere with probability one. I'm not sure if the second result is true.
If these results are indeed true, I'm also guessing that they hold even if $\mathbb{E}_X\left[\left\lVert f_n(X) \right\rVert^2\right]$ is replaced with $\mathbb{E}_X\left[\left\lVert f_n(X) \right\rVert\right]$ in their assumptions.
Context: Suppose I want to construct a nonparametric regression model between a dependent random variable $Y$ and an independent random variable $X$. Call the best regression model in the sense of minimum $L^2$ error $m(X) = \mathbb{E}[Y \mid X]$. Suppose we have iid data $\{(x_i,y_i)\}_{i=1}^{n}$, and we estimate a regression model $m_n(x)$ using some nonparametric regression technique.
Let $f_n(z) := m(z) - m_n(z)$. The first statement says that if the expected mean-squared error between the true model and the estimated model over $\Pi$ goes to zero almost surely, then the regression estimate is weakly consistent at almost every $X$. The second statement weakens the assumption to convergence in probability.