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It is known that sample quantiles from any distribution are Gaussian-distributed:

$\sqrt{n} (X_{[np]} - q_p) \sim \mathcal{N}(0, \frac{p(1-p)}{(F'(q_p))^2 })$

Here sample size is $n$, $X_{(np)}$ is the order statistic number $[np]$ (sorted ascending), $q_p$ is the real quantile of level $p$, $F$ is the distribution function. Order statistic $X_{(np)}$ serves as a sample quantile.

We also know that if we take sample quantiles of several levels (e.g. two levels, $p_1$ and $p_2$), they have a multivariate normal distribution with covariations $Cov(X_{[np_1]}, X_{[np_2]}) = \frac{p_1 (1-p_2)}{n F'(q_{p_1}) F'(q_{p_2})}$.

Does this multi-variate approach give rise to Guassian processes? If we consider $k$ sample quantiles, where $k \to \infty$, we can think of them as of a Gaussian random vector. With $k \to \infty$, this random vectors converge to a realisation of a Gaussian stochastic process with covariance matrix defined above.

Is this a valid approach to derive Gaussian processes, alternative to Bayesian kernel ridge regression?

References

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    $\begingroup$ What exactly are you trying to show here? Quantile function can be approximated by a Gaussian process, like any function, but I'm not sure what are you trying to take out of it? $\endgroup$ Commented Jul 11, 2023 at 11:16
  • $\begingroup$ @Tim Sample quantile is actually gaussian distributed if sample size $n \to \infty$. I am trying to establish a correspondence between Gaussian processes and multi-varaite sample quantile vector. I feel that Gaussian process is a generalization of the latter, but googling never mentions such an approach to their derivation. So, I am wondering, if you anyone ever encountered such an approach to defining Gaussian processes? $\endgroup$ Commented Jul 11, 2023 at 11:35
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    $\begingroup$ GP obviously isn't a generalization of the sample quantiles. For it to be, the covariance of sample quantiles should be able to take the form of an arbitrary kernel function. $\endgroup$ Commented Jul 11, 2023 at 11:58
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    $\begingroup$ Your general statement about the asymptotic distribution isn't always correct: you need $F$ to be differentiable in a neighborhood of the quantile. See stats.stackexchange.com/questions/45124. But that doesn't change your question. Notice, however, that the correlation coefficients in your process are always positive, whereas a general Gaussian process can have negative correlations. $\endgroup$ Commented Jul 11, 2023 at 13:57

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