Let $T:=[-1,1]^{n-1}\times (0,1]$. Let


$$f\_n(x\_1,\cdots,x\_n,w\_1,\cdots,w\_n):=g(x\_1,w\_1)+\cdots+g(x\_n,w\_n)=\sum\_{i=1}^ng(x\_i,w\_i),$$
where


(i) $w\_1,\cdots,w\_n$ are i.i.d. Gaussian random variables


(ii) $g(x\_i,w\_i)$ is a smooth function ($g\in C^\infty$)


(iii) $\mathbb{E}g(x\_i,w\_i)>0$


(iv) $\inf\_{(x\_1,\cdots,x\_n)\in T}\mathbb{E}(\sum\_{i=1}^ng(x\_i,w\_i))=0$


My question: is there any difference between the following two?


(1) for any $(x\_1,\cdots,x\_n)\in T$, we have $$P(\sum\_{i=1}^ng(x\_i,w\_i)>0)\rightarrow 1$$ when $n$ goes to infinity.


(2) $$P\big(\inf\_{(x\_1,\cdots,x\_n)\in T}\sum\_{i=1}^ng(x\_i,w\_i)>0\big)\rightarrow 1$$ when $n$ goes to infinity.


Thanks for any suggestion!



Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows
A trivial counterexample:
$$g(x,w):=x.$$


Then convergence (1) holds but convergence (2) does not.


On the other hand, of course, convergence (2) always implies convergence (1).


