Moment-generating function of Marchenko-Pastur?

Question

Suppose $h$ is distributed according to Marchenko-Pastur law with scale parameter $\sigma$. Is there a closed form expression or a good approximation for the following two quantities?

$$\text{mgf}(s)=E_h[\exp(h s)]$$

$$f(s)=E_h[h\exp(h s)]$$

Only negative values of $s$ need to be considered.

Motivation: these functions should describe expected error and expected loss curves for gradient descent on large linear problems as detailed in this paper by Paquette - https://cypaquette.github.io/Research/High_dimensional_optimization_survey.pdf

Answer 1

I'm not sure that the following gets you what you need for the mgf but here is one approach to obtain a more specific form of the mgf:

Mathematica can produce the moments in the following manner:

{#, Moment[MarchenkoPasturDistribution[λ, σ], #]} & /@ Range[9] // TableForm

Looking up the sequence of coefficients in oeis.org finds the formula for the $n$-th moment:

m[n_, σ_, λ_] := σ^(2 n) Sum[λ^(k - 1) Binomial[n - 1, k - 1] Binomial[n, k - 1]/k, {k, 1, n}]

Then construct the mgf with brute force:

mgf[s_, σ_, λ_, nmax_ : Infinity] := 1 + Sum[m[i, σ, λ] s^i/i!, {i, 1, nmax}]
mgf[s, σ, λ]

$$\sum _{i=1}^{\infty } \frac{s^i \sigma ^{2 i} \, _2F_1(1-i,-i;2;\lambda )}{i!}+1$$

For some values of $\sigma$ and $\lambda$ closed forms exist (again, in the eye of the beholder):

mgf[s, σ, 1]
(* E^(2 s σ^2) (BesselI[0, 2 s σ^2] - BesselI[1, 2 s σ^2]) *)

The same thing can be done with $f(s)$:

f[s_, σ_, λ_, nmax_ : Infinity] := Sum[m[i + 1, σ, λ] s^i/i!, {i, 0, nmax}]
f[s, σ, λ]

$$\sum _{i=0}^{\infty } \frac{s^i \sigma ^{2 (i+1)} \, _2F_1(-i-1,-i;2;\lambda )}{i!}$$

with

f[s, σ, 1]

resulting in $$\frac{e^{2 s \sigma ^2} I_1\left(2 s \sigma ^2\right)}{s}$$

where $I_n(z)$ is the modified Bessel function of the first kind.