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This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? 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 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.

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

This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? 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 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?

This question is partially inspired by following MO post: What are some of the surprising results of finite sample statistical estimation? 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 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.

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

This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? 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 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.

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