It is well known, due to the Markov inequality and the Borel-Cantelli lemma, that for a sequence of random variables $(X_n)_{n\in \mathbb{N}}$ that converge to $0$ in probability, from
$\sum E(|X_n|) < + \infty$ one can deduce that the convergence also happens almost surely.
Question: I am wondering if the same analogy can be made with a stochastic process $(Y_t)_{t \in \mathbb R^+}$ (possibly with continuous paths) converging to $0$ in probability, for example under the assumptions:
1) $\sum E(|Y_{n}|) < +\infty $, and
2) $\mathbb{R}^+\ni t \mapsto E(|Y_t|) $ is decreasing.
If not, is there any other assumption under which we could draw such a conclusion (i.e. that $Y_t \to 0$ a.s.)?
In the related question Is there a continuous version of the Borel-Cantelli lemma? the answer provided is not sufficient to conclude an a.s. convergence.
Best Answer
I was asking myself the same question, but under weaker assumptions and I think I finally found an answer. Unfortunately there is a counterexample satisfying even the strong assumptions above. It was motivated by the answer of VF1 for the question regarding the continuous version of the Borel-Cantelli lemma. Unfortunately his answer didn't satisfy the convergence in probability to $0$, so I took another function and modified the process. Please check if the couterexample is correct.
For the sake of simplicity we consider only the idex set $[3,\infty[$ instead of $\mathbb{R}_+$, but the example can be shifted of course.
Consider the test function (i.e. element of $C^\infty_c(\mathbb{R})$) $$\phi_b(x)=\begin{cases}\exp(b^2/(x^2-b^2),\qquad&x\in]-b,b[\\ 0,\qquad&x\notin]-b,b[ \end{cases}$$ for $b>0$ and the probability space $(\Omega, \sigma, P)=([0,1[,\mathcal{B}([0,1[),\lambda)$, whereas $\lambda$ is the Lebesgue measure. Note: $\forall b>0:\phi_b(0)=e^{-1}$ and $\forall n\in\mathbb{N}\forall \omega\in[0,1[:$ $$\phi_{1/n^2}(t-\omega-n)=\begin{cases}\exp(n^{-4}/((t-\omega-n)^2-n^{-4}),\qquad&t\in]n+\omega-n^{-2},n+\omega+n^{-2}[\\ 0,\qquad&t\notin]n+\omega-n^{-2},n+\omega+n^{-2}[. \end{cases}$$ We define the process $Y_t(\omega):=\sum_{n=1}^\infty \phi_{1/n^2}(t-\omega-n).$ Note, that $Y$ is continuous and for fixed $\omega$ there is for $t\geq3$ always at most one of the functions inside the sum unequal to $0$. Obviously $Y$ is not converging to $0$ a.s., since $\forall k\in\mathbb{N}:Y_{\omega+k}(\omega)=e^{-1}$. But it holds for all $\epsilon>0$ \begin{align} P(Y_t>\epsilon)\leq P(Y_t>0)=& P\left(\omega\in[0,1[: \sum_{n=1}^\infty \phi_{1/n^2}(t-\omega-n)>0\right)\\ \leq& 3P\left(\omega\in[0,1[: |t-\lfloor t\rfloor-\omega|\leq\frac{1}{(\lfloor t\rfloor-1)^2} \right)\\ \leq&3\lambda\left([t-\lfloor t\rfloor-\frac{1}{(\lfloor t\rfloor-1)^2},t-\lfloor t\rfloor+\frac{1}{(\lfloor t\rfloor-1)^2}]\right)\\ =&\frac{6}{(\lfloor t\rfloor-1)^2}, \end{align} so it is converging to $0$ in probability. Here in the second inequality we estimate with $\lfloor t\rfloor-1$ instead of $\lfloor t\rfloor$ or $\lfloor t\rfloor+1$, since for big $\omega$ and small $t-\lfloor t\rfloor$ the $t$ can be "in the bump belonging to the previous integer". Analogous for small $\omega$ and big $t-\lfloor t\rfloor$ with the bump of the following integer. The three possible cases cause the factor 3. Furthermore $$\int_3^\infty P(Y_t>\epsilon)\mathrm{d}t\leq\int_3^\infty \frac{6}{(\lfloor t\rfloor-1)^2}\mathrm{d}t=6\sum_{n=2}^{\infty}\frac{1}{n^2}<\infty $$ the continuous analogy of the Borel-Cantelli condition holds and $$\int_3^\infty E(Y_t)\mathrm{d}t=\int_3^\infty E(Y_t\cdot{1}_{Y_t>0})\mathrm{d}t\leq e^{-1}\int_3^\infty P(Y_t>0)\mathrm{d}t\leq e^{-1}\int_3^\infty \frac{6}{(\lfloor t\rfloor-1)^2}\mathrm{d}t<\infty $$ condition 1) is satisfied. Also condition 2) seems to me beeing satisfied, but it is a lot of calculation \begin{align} \frac{\mathrm{d}}{\mathrm{d}t} E(Y_t)=\frac{\mathrm{d}}{\mathrm{d}t}\int_0^1 \sum_{n=1}^\infty \phi_{1/n^2}(t-\omega-n) \mathrm{d}\lambda(\omega)=&\sum_{n=1}^\infty \int_0^1 \frac{\mathrm{d}}{\mathrm{d}t}\phi_{1/n^2}(t-\omega-n) \mathrm{d}\lambda(\omega)\\ =&\sum_{n=1}^\infty \int_0^1 -\frac{\mathrm{d}}{\mathrm{d}\omega}\phi_{1/n^2}(t-\omega-n) \mathrm{d}\lambda(\omega)\\ =&\sum_{n=1}^\infty \phi_{1/n^2}(t-0-n)-\phi_{1/n^2}(t-1-n)\\ =&\sum_{n=1}^\infty \phi_{1/n^2}(t-n)-\phi_{1/n^2}(t-(n+1)). \end{align} To see that this is less or equal to zero we consider \begin{align} &\sum_{n=1}^\infty \phi_{1/n^2}(t-n)-\phi_{1/n^2}(t-(n+1))\\ &=\phi_{1}(t-1)-\phi_{1}(t-2)+\phi_{1/4}(t-2)-\phi_{1/4}(t-3)+\phi_{1/9}(t-3)-...\\ &=\underbrace{\phi_{1}(t-1)}_{=0,\text{ für } t\geq2}+\underbrace{\big(\phi_{1/4}(t-2)-\phi_{1}(t-2)\big)}_{\leq0}+\underbrace{\big(\phi_{1/9}(t-3)-\phi_{1/4}(t-3)\big)}_{\leq0}+... \end{align} Since we consider only $t\geq3$, $\frac{\mathrm{d}}{\mathrm{d}t} E(Y_t)\leq0$ holds and condition 2) is satisfied.
In summary, we found a nonnegative process $Y$ with a.s. $C^\infty([3,\infty[)$ paths and
Furthermore, from 2. and 3. we can conclude