I am building a model based on Gaussian processes and want to assume something like as my sample size $n$ gets large my prediction error goes to 0. In other words,a re there any proofs or theorems out there that show something like 

$$Pr((\hat y -y)=0)\rightarrow1$$ as $n\rightarrow\infty$? Here $\hat y$ is my predicted value and $y$ is my true value.