To understand the challenges of high-dimensional statistics, it is essential to first recall the classical asymptotic results that form the foundation of traditional statistical inference. These results include consistency, asymptotic normality, and efficiency of estimators. Consistency means that as the sample size $n$ grows, an estimator converges in probability to the true parameter value. Asymptotic normality ensures that the scaled difference between the estimator and the true parameter converges in distribution to a normal distribution, enabling the use of confidence intervals and hypothesis tests. Efficiency refers to achieving the lowest possible variance among all unbiased estimators. However, these results are derived under explicit assumptions about the relationship between the number of parameters $p$ and the sample size $n$. Specifically, classical theory requires that $p$ remains fixed or grows much more slowly than $n$ as $n$ increases. When $p$ becomes large relative to $n$, these guarantees can no longer be taken for granted.

As you move into the high-dimensional regime, where the ratio $p/n$ is no longer negligible, classical asymptotic theory encounters fundamental limitations. A key concept here is the phase transition: a threshold phenomenon where the qualitative behavior of statistical estimators abruptly changes as $p/n$ crosses certain critical values. For instance, the sample covariance matrix is invertible and well-behaved only when $p < n$; as soon as $p$ approaches or exceeds $n$, inversion becomes impossible and many estimators become ill-posed. This threshold marks a breakdown of classical results—consistency, normality, and efficiency may all fail beyond this point. More generally, as $p/n$ grows, estimators can become unstable, confidence intervals may no longer be valid, and the power of hypothesis tests can collapse. These phase transitions highlight the need to carefully analyze the scaling of $p$ with respect to $n$ in high-dimensional settings.

The breakdown of classical results in high dimensions is not just an algebraic artifact, but is deeply rooted in the geometry of high-dimensional spaces. As dimensionality increases, geometric phenomena such as concentration of measure and volume collapse become dominant. For example, in high dimensions, most points in a sphere concentrate near the surface, and the volume of the unit ball shrinks rapidly relative to its enclosing cube. These effects mean that distances between points become less informative, projections can become nearly orthogonal, and the intuition developed in low dimensions fails. Such geometric properties undermine the assumptions behind classical statistical estimators and explain why their behavior changes so dramatically as $p/n$ increases.

Given these challenges, it is clear that the classical asymptotic toolkit is insufficient for high-dimensional inference. The failure of consistency, normality, and efficiency as $p/n$ grows necessitates the development of new theoretical frameworks and methodologies. High-dimensional statistics requires alternative tools that explicitly account for the scaling of $p$ and the unique geometry of high-dimensional spaces. This shift has led to the emergence of concepts such as sparsity, regularization, and random matrix theory, which are designed to handle the complexities of modern data analysis where $p$ can be comparable to, or even larger than, $n$.