This question is partially inspired by the following MO post: https://mathoverflow.net/questions/261643/what-are-some-of-the-surprising-results-of-finite-sample-statistical-estimation/262341#262341 and current heated research front of high dimensional statistics.

Instead of asking about surprising results in high dimensions, I will ask **what kind of results that holds in low dimensions fails to hold in a higher dimension. And what is its relation with other mathematical branch?**

(By *high dimensional statistics* we usually refer to a high dimension covariate space instead of response space. For example, in a regression setting $Y=f(X)$ we tend to say a problem is of high dimension if $\dim\mathcal{X}\gg \dim\mathcal{Y}$)

For one simplest example, we know that [James-Stein estimator][1] performs better than maximum likelihood estimator in terms of $L^2$ norm when the dimension $\dim\mathcal{X}=d\geq 3$; and it turns out to be an *equivalent statement* that a symmetric random walk in $\mathcal{X}=\mathbb{R}^d$ is transient when $d\geq 3$ via an infinitely divisible stochastic process. Another example is provided in the answer below.

Are there other such examples that can relate high dimensional phenomena in statistics?


  [1]: https://mathoverflow.net/questions/93745/james-stein-phenomenon-what-does-it-mean-that-a-james-stein-estimator-beats-lea/266937#266937