# Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows

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!

• What is $\vec x^*$? Commented May 28, 2023 at 1:49
• "When $n$ goes to infinity"? So you have a whole family of functions $f$, one for each $n$? Commented May 28, 2023 at 7:28
• @IosifPinelis sorry for the confusion, I re-edited my question. I hope this time is much clear. Commented May 28, 2023 at 9:07
• @GeraldEdgar yes, $f$ is a random function $X_{t\in T\subset R^n}$ and its dimension $n$ grows Commented May 28, 2023 at 9:44
• What could $X_{t\in T}$ possibly mean? Commented May 28, 2023 at 13:49

A trivial counterexample: $$g(x,w):=x.$$
• thank you! Other than '$T$ is compact' (then $\inf=\min$), is there other condition that we could add such that (1) implies (2)? Commented May 28, 2023 at 18:18