The Stieltjes Transform

The Moment Generating Function of Spectral Theory

The Stieltjes transform plays the same role for spectral distributions that the moment generating function plays for ordinary probability distributions: it encodes all the information about the distribution in a single analytic function. For the Marchenko-Pastur law, the Stieltjes transform satisfies a simple quadratic equation β€” and this equation is the key to computing ergodic MIMO capacity in closed form.

Definition:

The Stieltjes Transform

Let FF be a probability distribution function on [0,∞)[0, \infty). The Stieltjes transform of FF is the function mF(z)=∫0∞1Ξ»βˆ’z dF(Ξ»),z∈Cβˆ–supp(F).m_F(z) = \int_0^{\infty} \frac{1}{\lambda - z}\, dF(\lambda), \quad z \in \mathbb{C} \setminus \text{supp}(F). For z∈C+={z:Im(z)>0}z \in \mathbb{C}^+ = \{z : \text{Im}(z) > 0\}, mF(z)m_F(z) is well-defined and analytic, with Im(mF(z))<0\text{Im}(m_F(z)) < 0 (i.e., mFm_F maps the upper half-plane to the lower half-plane).

The Stieltjes transform determines FF uniquely via the inversion formula: f(Ξ»)=βˆ’1Ο€lim⁑η↓0Im mF(Ξ»+jΞ·)f(\lambda) = -\frac{1}{\pi}\lim_{\eta \downarrow 0} \text{Im}\, m_F(\lambda + j\eta). This means we can recover the density from the Stieltjes transform by taking the imaginary part on the real axis.

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Theorem: Stieltjes Inversion Formula

If FF has a continuous density ff at the point Ξ»\lambda, then f(Ξ»)=βˆ’1Ο€lim⁑η↓0Im mF(Ξ»+jΞ·).f(\lambda) = -\frac{1}{\pi}\lim_{\eta \downarrow 0} \text{Im}\, m_F(\lambda + j\eta). More generally, for any continuity interval (a,b)(a, b) of FF: F(b)βˆ’F(a)=βˆ’1Ο€lim⁑η↓0∫abIm mF(x+jΞ·) dx.F(b) - F(a) = -\frac{1}{\pi}\lim_{\eta \downarrow 0} \int_a^b \text{Im}\, m_F(x + j\eta)\, dx.

The Stieltjes transform mF(z)m_F(z) is smooth for zz away from the real axis. As zz approaches the real axis (i.e., Ξ·β†’0\eta \to 0), the imaginary part of mFm_F "reveals" the density: where the density is large, ∣Im mF∣|\text{Im}\, m_F| is large; where the density is zero, Im mFβ†’0\text{Im}\, m_F \to 0.

Proof
Compute the imaginary part

For z=Ξ»+jΞ·z = \lambda + j\eta with Ξ·>0\eta > 0: Im mF(z)=βˆ«βˆ’Ξ·(tβˆ’Ξ»)2+Ξ·2 dF(t).\text{Im}\, m_F(z) = \int \frac{-\eta}{(t - \lambda)^2 + \eta^2}\, dF(t). The kernel Ξ·Ο€[(tβˆ’Ξ»)2+Ξ·2]\frac{\eta}{\pi[(t-\lambda)^2 + \eta^2]} is the Cauchy (Lorentzian) density centered at Ξ»\lambda with scale Ξ·\eta. As Ξ·β†’0\eta \to 0, it converges to Ξ΄(tβˆ’Ξ»)\delta(t - \lambda).

Take the limit

By the approximation-to-identity property: βˆ’1Ο€Im mF(Ξ»+jΞ·)=∫1πη(tβˆ’Ξ»)2+Ξ·2 dF(t)β†’f(Ξ»)-\frac{1}{\pi}\text{Im}\, m_F(\lambda + j\eta) = \int \frac{1}{\pi}\frac{\eta}{(t-\lambda)^2 + \eta^2}\, dF(t) \to f(\lambda) as Ξ·β†’0\eta \to 0, whenever ff is continuous at Ξ»\lambda.

Definition:

Stieltjes Transform of the Empirical Spectral Distribution

For an nΓ—nn \times n Hermitian matrix A\mathbf{A} with eigenvalues Ξ»1,…,Ξ»n\lambda_1, \ldots, \lambda_n, the Stieltjes transform of its ESD is mFA(z)=1nβˆ‘i=1n1Ξ»iβˆ’z=1ntr(Aβˆ’zI)βˆ’1.m_{F^{\mathbf{A}}}(z) = \frac{1}{n}\sum_{i=1}^n \frac{1}{\lambda_i - z} = \frac{1}{n}\text{tr}(\mathbf{A} - z\mathbf{I})^{-1}. The key insight: the Stieltjes transform of the ESD is the normalized trace of the resolvent (Aβˆ’zI)βˆ’1(\mathbf{A} - z\mathbf{I})^{-1}.

Theorem: Fixed-Point Equation for the Marchenko-Pastur Stieltjes Transform

The Stieltjes transform m(z)=mFΞ²(z)m(z) = m_{F_\beta}(z) of the Marchenko-Pastur distribution satisfies the fixed-point equation m(z)=11βˆ’Ξ²βˆ’zβˆ’Ξ²z m(z),z∈C+.m(z) = \frac{1}{1 - \beta - z - \beta z\, m(z)}, \quad z \in \mathbb{C}^+. Equivalently, m(z)m(z) is the unique solution in Cβˆ’\mathbb{C}^- (negative imaginary part) of the quadratic Ξ²z m2+(zβˆ’1+Ξ²) m+1=0.\beta z\, m^2 + (z - 1 + \beta)\, m + 1 = 0.

The fixed-point structure arises because the resolvent of a Wishart matrix 1mHHH\frac{1}{m}\mathbf{H}^H\mathbf{H} satisfies a self-consistent equation: each row of H\mathbf{H} contributes to the resolvent, but its contribution depends on the resolvent of the matrix with that row removed. In the large-dimensional limit, removing one row has negligible effect, and the equation becomes deterministic.

Proof
Matrix identity

Write W=1mHHH\mathbf{W} = \frac{1}{m}\mathbf{H}^H\mathbf{H} and consider the resolvent Q(z)=(Wβˆ’zI)βˆ’1\mathbf{Q}(z) = (\mathbf{W} - z\mathbf{I})^{-1}. Using the matrix inversion lemma and the rank-1 structure of each row's contribution: 1mtr Q(z)=1mβˆ‘i=1n11mhiHQβˆ’i(z)hiβˆ’z+correction,\frac{1}{m}\text{tr}\,\mathbf{Q}(z) = \frac{1}{m}\sum_{i=1}^n \frac{1}{\frac{1}{m}\mathbf{h}_i^H \mathbf{Q}_{-i}(z) \mathbf{h}_i - z + \text{correction}}, where hi\mathbf{h}_i is the ii-th row of H\mathbf{H} and Qβˆ’i\mathbf{Q}_{-i} is the resolvent with row ii removed.

Self-consistency in the limit

In the limit, 1mhiHQβˆ’i(z)hiβ†’1mtr Q(z)=m(z)\frac{1}{m}\mathbf{h}_i^H \mathbf{Q}_{-i}(z) \mathbf{h}_i \to \frac{1}{m}\text{tr}\,\mathbf{Q}(z) = m(z) by the law of large numbers (since hi\mathbf{h}_i has i.i.d. entries independent of Qβˆ’i\mathbf{Q}_{-i}). Substituting and summing over the nn rows: m(z)=n/mβˆ’z+n/mβ‹…(βˆ’zm(z))βˆ’1β‹…(terms)=11βˆ’Ξ²βˆ’zβˆ’Ξ²zm(z).m(z) = \frac{n/m}{-z + n/m \cdot (-z m(z))^{-1} \cdot (\text{terms})} = \frac{1}{1 - \beta - z - \beta z m(z)}.

Select the correct root

The quadratic Ξ²zm2+(zβˆ’1+Ξ²)m+1=0\beta z m^2 + (z - 1 + \beta) m + 1 = 0 has two roots. The correct one satisfies Im(m(z))<0\text{Im}(m(z)) < 0 for Im(z)>0\text{Im}(z) > 0 (the Stieltjes transform maps C+\mathbb{C}^+ to Cβˆ’\mathbb{C}^-). This uniquely determines: m(z)=βˆ’(zβˆ’1+Ξ²)+(zβˆ’1+Ξ²)2βˆ’4Ξ²z2Ξ²z,m(z) = \frac{-(z - 1 + \beta) + \sqrt{(z - 1 + \beta)^2 - 4\beta z}}{2\beta z}, where the branch of the square root is chosen to give Im(m)<0\text{Im}(m) < 0.

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Example: Ergodic MIMO Capacity via Stieltjes Transform

Use the Stieltjes transform to express the per-antenna ergodic capacity of an i.i.d. Rayleigh MIMO channel in terms of the Marchenko-Pastur Stieltjes transform. Show that the result involves only a one-dimensional integral (or a fixed-point equation), not a matrix-dimensional Monte Carlo average.

Solution
Capacity as a log-det integral

The per-antenna capacity is CΛ‰=∫0∞log⁑2(1+ρλ) dFΞ²(Ξ»).\bar{C} = \int_0^{\infty} \log_2(1 + \rho\lambda)\, dF_\beta(\lambda).

Relate to the Stieltjes transform

Using integration by parts and the identity ddρ∫log⁑(1+ρλ) dF(Ξ»)=∫λ1+ρλ dF(Ξ»)\frac{d}{d\rho}\int \log(1 + \rho\lambda)\, dF(\lambda) = \int \frac{\lambda}{1 + \rho\lambda}\, dF(\lambda), we recognize the right side as related to mF(βˆ’1/ρ)m_F(-1/\rho): ∫λ1+ρλ dF(Ξ»)=βˆ’1Οβˆ’1ρmF(βˆ’1/ρ).\int \frac{\lambda}{1 + \rho\lambda}\, dF(\lambda) = -\frac{1}{\rho} - \frac{1}{\rho} m_F(-1/\rho).

Final formula

Integrating over ρ\rho from 00 to the target SNR: CΛ‰=1ln⁑2∫0ρ(βˆ’1tβˆ’1tmFΞ²(βˆ’1/t))dt.\bar{C} = \frac{1}{\ln 2}\int_0^{\rho} \left(-\frac{1}{t} - \frac{1}{t}m_{F_\beta}(-1/t)\right) dt. Since mFΞ²m_{F_\beta} satisfies the quadratic fixed-point equation, this integral can be evaluated analytically. The result (due to Telatar) is a closed-form expression involving logarithms and the parameters Ξ²\beta and ρ\rho.

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Recovering the MP Density from Im mF(Ξ»+jΞ·)\text{Im}\, m_F(\lambda + j\eta)

Visualize the Stieltjes inversion formula: as Ξ·β†’0\eta \to 0, the imaginary part of mF(Ξ»+jΞ·)m_F(\lambda + j\eta) converges to βˆ’Ο€fMP(Ξ»)-\pi f_{\mathrm{MP}}(\lambda).

Parameters
0.5
0.05

Ergodic MIMO Capacity: RMT vs. Monte Carlo

Compare the asymptotic capacity formula (computed via the Stieltjes transform of the Marchenko-Pastur law) against Monte Carlo simulation for finite-dimensional MIMO channels.

Parameters
32
64
10

Quick Check

The Stieltjes transform mF(z)m_F(z) of a probability distribution on [0,∞)[0,\infty) maps the upper half-plane C+\mathbb{C}^+ to which region?

The lower half-plane Cβˆ’\mathbb{C}^-

The upper half-plane C+\mathbb{C}^+

The real line R\mathbb{R}

The unit disk

Correction:
The lower half-plane Cβˆ’\mathbb{C}^-

Since Im(1/(Ξ»βˆ’z))<0\text{Im}(1/(\lambda - z)) < 0 for Im(z)>0\text{Im}(z) > 0 and λ∈R\lambda \in \mathbb{R}, the integral inherits the sign.

Computing the MP Stieltjes Transform via Fixed-Point Iteration

Complexity: O(K)O(K) where KK is the number of iterations (typically K<30K < 30)
Input: z∈C+z \in \mathbb{C}^+, aspect ratio β>0\beta > 0, tolerance ϡ>0\epsilon > 0
Output: m(z)β‰ˆmFΞ²(z)m(z) \approx m_{F_\beta}(z)
1. Initialize m(0)β†βˆ’1/zm^{(0)} \leftarrow -1/z
2. for k=0,1,2,…k = 0, 1, 2, \ldots do
3. m(k+1)←11βˆ’Ξ²βˆ’zβˆ’Ξ²z m(k)\quad m^{(k+1)} \leftarrow \frac{1}{1 - \beta - z - \beta z\, m^{(k)}}
4. \quad if ∣m(k+1)βˆ’m(k)∣<Ο΅|m^{(k+1)} - m^{(k)}| < \epsilon then break
5. end for
6. return m(k+1)m^{(k+1)}

The fixed-point iteration converges rapidly because the map is a contraction in the upper half-plane. For zz on the real axis (inside the support), use z=Ξ»+jΞ·z = \lambda + j\eta with small Ξ·>0\eta > 0 and take the imaginary part to recover the density.

Why the Stieltjes Transform is So Useful

The Stieltjes transform has three properties that make it the tool of choice for random matrix theory:

  1. Inversion: the density is recovered from Im mF(Ξ»+jΞ·)\text{Im}\, m_F(\lambda + j\eta).
  2. Fixed-point equations: for structured random matrices (Wishart, correlated, sums), the Stieltjes transform of the LSD satisfies a finite-dimensional equation, even though the underlying matrices have growing dimension.
  3. Functionals: quantities like ∫log⁑(1+ρλ) dF(Ξ»)\int \log(1 + \rho\lambda)\, dF(\lambda) (capacity) or ∫λ(1+ρλ)2 dF(Ξ»)\int \frac{\lambda}{(1+\rho\lambda)^2}\, dF(\lambda) (MMSE) can be expressed as simple functions of mFm_F evaluated at specific points.
πŸ”§Engineering Note

Numerical Evaluation of the Stieltjes Transform

When evaluating mF(Ξ»+jΞ·)m_F(\lambda + j\eta) numerically to recover the density, the choice of Ξ·\eta matters: too large and the density is smoothed out (losing the sharp edges at λ±\lambda_\pm); too small and numerical issues arise. A practical choice is η∼0.01β‹…(Ξ»+βˆ’Ξ»βˆ’)\eta \sim 0.01 \cdot (\lambda_+ - \lambda_-). For the fixed-point iteration, convergence is guaranteed for z∈C+z \in \mathbb{C}^+ but may be slow near the edge of the support. The direct quadratic formula solution is preferred when available.

Practical Constraints
  • β€’

    Typical Ξ·\eta range: 10βˆ’310^{-3} to 10βˆ’110^{-1} times the support width

  • β€’

    Fixed-point iteration: 10--30 iterations suffice for 10βˆ’1010^{-10} accuracy

Stieltjes Transform

The function mF(z)=∫(Ξ»βˆ’z)βˆ’1 dF(Ξ»)m_F(z) = \int (\lambda - z)^{-1}\, dF(\lambda) for z∈Cβˆ–supp(F)z \in \mathbb{C} \setminus \text{supp}(F). It uniquely determines FF and is the main analytic tool for characterizing limiting spectral distributions in random matrix theory.

Related: The Stieltjes Transform

Resolvent

For a matrix A\mathbf{A} and complex zβˆ‰spec(A)z \notin \text{spec}(\mathbf{A}), the resolvent is Q(z)=(Aβˆ’zI)βˆ’1\mathbf{Q}(z) = (\mathbf{A} - z\mathbf{I})^{-1}. The normalized trace of the resolvent gives the Stieltjes transform of the ESD.

Related: Stieltjes Transform of the Empirical Spectral Distribution

Key Takeaway

The Stieltjes transform converts the eigenvalue distribution problem into an analytic function problem. For the Marchenko-Pastur law, it satisfies a simple quadratic equation, enabling closed-form computation of MIMO capacity and related functionals without Monte Carlo simulation.

The Marchenko-Pastur LawDeterministic Equivalents