Proof of Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits

Each result cited is universally quantified over the data in its own statement.

Preliminaries. Let $\nu\in\Sigma_{d}$. By The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns and The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes, $\mathcal{H}_{\nu}$ is the complex Hilbert completion of $(\mathcal{P}_{d},h_{\nu})$ with $h_{\nu}(p,q)=\nu(p^{*}q)$, and $\widehat{p}=J_{h_{\nu}}(p)$. By The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert it is a complex Hilbert space, so its pairing satisfies conditions 1 to 4 of Complex Inner Product Space; in particular $\langle z\xi,\eta\rangle=\overline{z}\langle\xi,\eta\rangle$ and $\langle\xi+\xi',\eta\rangle=\langle\xi,\eta\rangle+\langle\xi',\eta\rangle$ (conditions 1 to 3 together with claim 1 of Properties of Complex Conjugation and Modulus), and $|\langle\xi,\eta\rangle|\le\lVert\xi\rVert\,\lVert\eta\rVert$ by claim 1 of The Induced Norm is a Norm, and Induces a Metric. By The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, $p\mapsto\widehat{p}$ is complex-linear and

$$\langle\widehat{p},\widehat{q}\rangle_{\mathcal{H}_{\nu}}=\nu(p^{*}q)\qquad(p,q\in\mathcal{P}_{d}),\qquad\text{(G)}$$

so $\lVert\widehat{p}\rVert^{2}_{\mathcal{H}_{\nu}}=\nu(p^{*}p)=\lVert p\rVert_{\nu}^{2}$ and, both being nonnegative square roots of $\nu(p^{*}p)$ (Existence and Uniqueness of the Nonnegative Square Root), $\lVert\widehat{p}\rVert_{\mathcal{H}_{\nu}}=\lVert p\rVert_{\nu}$. For $j\in[d]$ the variable $x_{j}$ is the monomial of the one-letter word $(j)$, whose only position is $l=1$ with $(j)_{<1}=(j)_{>1}=\varnothing$; so by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient and condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state,

$$\partial^{\nu}_{j}(x_{j})=\delta_{jj}\,\nu(1)\,\nu(1)=1.\qquad\text{(D)}$$

Also $x_{j}^{*}=x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint. Recall that $d\in\mathbb{N}=\{1,2,\dots\}$ by Natural Numbers, so $d\ge1$.

Clause 1 (Cramer-Rao). Let $\xi_{\lambda}=(\xi_{1},\dots,\xi_{d})$ be the conjugate variables of $\lambda$. For $j\in[d]$, The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate and (D) give $\langle\xi_{j},\widehat{x_{j}}\rangle=\partial^{\lambda}_{j}(x_{j})=1$, and (G) gives $\lVert\widehat{x_{j}}\rVert^{2}=\lambda(x_{j}^{*}x_{j})=\lambda(x_{j}^{2})$. By Cauchy-Schwarz Inequality in a Complex Inner Product Space,

$$1=|\langle\xi_{j},\widehat{x_{j}}\rangle|^{2}\le\lVert\xi_{j}\rVert^{2}\,\lambda(x_{j}^{2}).$$

Hence $\lambda(x_{j}^{2})\neq0$, and since $\lambda(x_{j}^{2})\ge0$ we get $\lambda(x_{j}^{2})>0$ for every $j$; as $d\ge1$, $M(\lambda)=\sum_{j=1}^{d}\lambda(x_{j}^{2})>0$ (Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws). Moreover $1\le\bigl(\lVert\xi_{j}\rVert\,\lVert\widehat{x_{j}}\rVert\bigr)^{2}$ with $\lVert\xi_{j}\rVert\,\lVert\widehat{x_{j}}\rVert\ge0$ forces $1\le\lVert\xi_{j}\rVert\,\lVert\widehat{x_{j}}\rVert$ (if the product $t$ satisfied $0\le t<1$ then $t^{2}<1$). Let $X=(\lVert\xi_{1}\rVert,\dots,\lVert\xi_{d}\rVert)$ and $Y=(\lVert\widehat{x_{1}}\rVert,\dots,\lVert\widehat{x_{d}}\rVert)$, points of $\mathbb{R}^{d}$. Summing over $j$ and using the dot product and Cauchy-Schwarz Inequality for the Euclidean Dot Product,

$$d=\sum_{j=1}^{d}1\le\sum_{j=1}^{d}X_{j}Y_{j}=X\cdot Y\le|X\cdot Y|\le\lVert X\rVert\,\lVert Y\rVert .$$

By Euclidean Norm on $\mathbb{R}^n$, $\lVert X\rVert^{2}=\sum_{j}\lVert\xi_{j}\rVert^{2}=\Phi^{*}(\lambda)$ by The Free Fisher Information of a Noncommutative Law §fisher, and $\lVert Y\rVert^{2}=\sum_{j}\lambda(x_{j}^{2})=M(\lambda)$. Since $0\le d\le\lVert X\rVert\,\lVert Y\rVert$, squaring preserves the inequality, and $d^{2}\le\lVert X\rVert^{2}\lVert Y\rVert^{2}=\Phi^{*}(\lambda)\,M(\lambda)$.

Clause 2 (Semicircular laws). Fix a real $s>0$, and let $T_{s}=(A,0)$ with $A_{ij}=\sqrt{s}\,\delta_{ij}$ be the affine datum from $d$ to $d$ variables of The Semicircular Law and Its Scalings §scaled, so that $\mathrm{sc}_{d,s}=\mathrm{sc}_{d}\circ\sigma_{T_{s}}\in\Sigma_{d}$. By Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple, the $j$-th entry of the tuple of $T_{s}$ is $0\cdot1+\sum_{l=1}^{d}A_{jl}x_{l}=\sqrt{s}\,x_{j}$, so $\sigma_{T_{s}}(x_{j})=\sqrt{s}\,x_{j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. By The Semicircular Law and Its Scalings §standard, $\mathrm{sc}_{d}$ satisfies the Schwinger-Dyson equation: $\mathrm{sc}_{d}(x_{j}q)=\partial^{\mathrm{sc}_{d}}_{j}(q)$ for $j\in[d]$ and $q\in\mathcal{P}_{d}$.

Let $j\in[d]$ and $p\in\mathcal{P}_{d}$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism, the bilinearity of the product (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra), linearity of $\mathrm{sc}_{d}$, the Schwinger-Dyson equation with $q=\sigma_{T_{s}}(p)$, and The Chain Rule for Free Difference Quotients under an Affine Substitution §chain-rule (with $m=n=d$, $T=T_{s}$, $\gamma=\mathrm{sc}_{d}$, $\mu=\mathrm{sc}_{d,s}$ and $k=j$),

$$\mathrm{sc}_{d,s}(x_{j}p)=\mathrm{sc}_{d}\bigl(\sqrt{s}\,x_{j}\,\sigma_{T_{s}}(p)\bigr)=\sqrt{s}\,\partial^{\mathrm{sc}_{d}}_{j}\bigl(\sigma_{T_{s}}(p)\bigr)=\sqrt{s}\sum_{i=1}^{d}A_{ij}\,\partial^{\mathrm{sc}_{d,s}}_{i}(p)=s\,\partial^{\mathrm{sc}_{d,s}}_{j}(p),\qquad\text{(S)}$$

using $\sqrt{s}\sqrt{s}=s$ (Existence and Uniqueness of the Nonnegative Square Root). Put $\xi_{j}=s^{-1}\widehat{x_{j}}\in\mathcal{H}_{\mathrm{sc}_{d,s}}$. Since $s^{-1}$ is real, $\overline{s^{-1}}=s^{-1}$, so by the Preliminaries, (G) with $\nu=\mathrm{sc}_{d,s}$, $x_{j}^{*}=x_{j}$ and (S),

$$\langle\xi_{j},\widehat{p}\rangle=s^{-1}\,\mathrm{sc}_{d,s}(x_{j}p)=\partial^{\mathrm{sc}_{d,s}}_{j}(p).$$

Thus $\mathrm{sc}_{d,s}$ has conjugate variables in the sense of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate, and by the uniqueness stated there they are $(s^{-1}\widehat{x_{1}},\dots,s^{-1}\widehat{x_{d}})$. Finally, (S) with $p=x_{j}$ and (D) with $\nu=\mathrm{sc}_{d,s}$ give $\mathrm{sc}_{d,s}(x_{j}x_{j})=s$, so by (G) $\lVert\xi_{j}\rVert^{2}=s^{-2}\,\mathrm{sc}_{d,s}(x_{j}x_{j})=s^{-1}$, and The Free Fisher Information of a Noncommutative Law §fisher gives $\Phi^{*}(\mathrm{sc}_{d,s})=\sum_{j=1}^{d}s^{-1}=d/s$.

Clause 3 (Closedness), Step 1: a uniform joint bound. Fix $k\in\mathbb{N}$ and let $(\xi^{k}_{1},\dots,\xi^{k}_{d})$ be the conjugate variables of $\lambda_{k}$, so $\sum_{j}\lVert\xi^{k}_{j}\rVert^{2}=\Phi^{*}(\lambda_{k})\le c$ by The Free Fisher Information of a Noncommutative Law §fisher. Let $p_{1},\dots,p_{d}\in\mathcal{P}_{d}$. By The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate, claim 7 of Properties of Complex Conjugation and Modulus, claim 1 of The Induced Norm is a Norm, and Induces a Metric with $\lVert\widehat{p_{j}}\rVert=\lVert p_{j}\rVert_{\lambda_{k}}$ from the Preliminaries, and Cauchy-Schwarz Inequality for the Euclidean Dot Product applied to the points $(\lVert\xi^{k}_{j}\rVert)_{j}$ and $(\lVert p_{j}\rVert_{\lambda_{k}})_{j}$ of $\mathbb{R}^{d}$,

$$\Bigl|\sum_{j=1}^{d}\partial^{\lambda_{k}}_{j}(p_{j})\Bigr|=\Bigl|\sum_{j=1}^{d}\langle\xi^{k}_{j},\widehat{p_{j}}\rangle\Bigr|\le\sum_{j=1}^{d}\lVert\xi^{k}_{j}\rVert\,\lVert p_{j}\rVert_{\lambda_{k}}\le\Bigl(\sum_{j=1}^{d}\lVert\xi^{k}_{j}\rVert^{2}\Bigr)^{1/2}\Bigl(\sum_{j=1}^{d}\lVert p_{j}\rVert^{2}_{\lambda_{k}}\Bigr)^{1/2},$$

where $(\cdot)^{1/2}$ is the nonnegative square root. Squaring both nonnegative sides,

$$\Bigl|\sum_{j=1}^{d}\partial^{\lambda_{k}}_{j}(p_{j})\Bigr|^{2}\le c\sum_{j=1}^{d}\lambda_{k}(p_{j}^{*}p_{j})\qquad(k\in\mathbb{N}).\qquad\text{(J}_{k}\text{)}$$

Step 2: passage to the limit. Keep $p_{1},\dots,p_{d}$ fixed. Call a sequence $(z_{k})$ of complex numbers convergent to $z$ if $(\operatorname{Re}z_{k})$ and $(\operatorname{Im}z_{k})$ converge to $\operatorname{Re}z$ and $\operatorname{Im}z$ (limits as in Limit of a Sequence of Real Numbers). Writing $\operatorname{Re}(zz')=\operatorname{Re}z\operatorname{Re}z'-\operatorname{Im}z\operatorname{Im}z'$, $\operatorname{Im}(zz')=\operatorname{Re}z\operatorname{Im}z'+\operatorname{Im}z\operatorname{Re}z'$ and taking real and imaginary parts of sums termwise (Real and Imaginary Parts of a Complex Number), Arithmetic of Limits of Real Sequences (claims 1 to 3) shows that sums, products and multiples by a fixed complex number of convergent complex sequences converge to the corresponding sums, products and multiples of the limits. By Weak-Star Convergence of Noncommutative Laws §weak-star, $(\lambda_{k}(x_{u}))_{k}$ converges to $\lambda(x_{u})$ for every word $u$. For $\nu\in\Sigma_{d}$, $j\in[d]$ and $0\neq p\in\mathcal{P}_{d}$, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, part (a), applied to $\partial^{\nu}_{j}$ gives

$$\partial^{\nu}_{j}(p)=\sum_{w\in\operatorname{supp}p}p(w)\sum_{l=1}^{|w|}\delta_{w_{l}j}\,\nu(x_{w_{<l}})\,\nu(x_{w_{>l}})$$

(with $|w|$ the length of $w$ and the inner sum $0$ for $w=\varnothing$), a finite sum whose coefficients do not depend on $\nu$; and $\partial^{\nu}_{j}(0)=0$. Hence $z_{k}=\sum_{j}\partial^{\lambda_{k}}_{j}(p_{j})$ converges to $z=\sum_{j}\partial^{\lambda}_{j}(p_{j})$, and by Modulus of a Complex Number and claims 1 and 2 of Arithmetic of Limits of Real Sequences, $|z_{k}|^{2}=(\operatorname{Re}z_{k})^{2}+(\operatorname{Im}z_{k})^{2}$ converges to $|z|^{2}$. Likewise $\lambda_{k}(p_{j}^{*}p_{j})$ is real, so it equals $\operatorname{Re}\lambda_{k}(p_{j}^{*}p_{j})$, which converges to $\operatorname{Re}\lambda(p_{j}^{*}p_{j})=\lambda(p_{j}^{*}p_{j})$; so the right side of (J$_{k}$) converges to $c\sum_{j}\lambda(p_{j}^{*}p_{j})$. Claim 1 of Order Properties of Limits of Real Sequences now gives, for all $p_{1},\dots,p_{d}\in\mathcal{P}_{d}$,

$$\Bigl|\sum_{j=1}^{d}\partial^{\lambda}_{j}(p_{j})\Bigr|^{2}\le c\sum_{j=1}^{d}\lVert p_{j}\rVert^{2}_{\lambda}.\qquad\text{(J)}$$

Step 3: the conjugate variables of $\lambda$. Fix $j\in[d]$. Taking $p_{i}=0$ for $i\neq j$ in (J) (note $\partial^{\lambda}_{i}(0)=0$ and $\lambda(0)=0$) and taking nonnegative square roots, $|\partial^{\lambda}_{j}(p)|\le c^{1/2}\lVert p\rVert_{\lambda}$ for every $p\in\mathcal{P}_{d}$. Let $\beta=\operatorname{Re}h_{\lambda}$ and let $H_{\beta}$ be the real Hilbert completion of $(\mathcal{P}_{d},\beta)$ with pairing $\langle\cdot,\cdot\rangle_{H_{\beta}}$, norm $|\cdot|$ and canonical map $J_{\beta}$, as in the preamble of The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form; there $\mathcal{P}_{d}$ is a real vector space by restriction of scalars. By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion and The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §canonical-map, $\mathcal{H}_{\lambda}=H_{\beta}$ as sets, with the same addition and real scalar multiplication, and $\widehat{p}=J_{\beta}(p)$; by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert the norm of $\mathcal{H}_{\lambda}$ is $|\cdot|$; and by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §hilbert, $H_{\beta}$ is a real Hilbert space. For $p\in\mathcal{P}_{d}$, $\lambda(p^{*}p)$ is real, so the seminorm of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences is $\lVert p\rVert_{\beta}=\sqrt{\operatorname{Re}\lambda(p^{*}p)}=\lVert p\rVert_{\lambda}$. Let $\Omega=J_{\beta}(1)$; by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §isometry, $|\Omega|^{2}=\langle\Omega,\Omega\rangle_{H_{\beta}}=\beta(1,1)=\operatorname{Re}\lambda(1^{*}1)=1$.

Define $T:\mathcal{P}_{d}\to H_{\beta}$ by $T(p)=\bigl(\operatorname{Re}\partial^{\lambda}_{j}(p)\bigr)\,\Omega$. It is linear over $\mathbb{R}$, because $\partial^{\lambda}_{j}$ is complex-linear and $\operatorname{Re}$ is additive and real-homogeneous, and by claim 6 of Properties of Complex Conjugation and Modulus $|T(p)|=|\operatorname{Re}\partial^{\lambda}_{j}(p)|\,|\Omega|\le|\partial^{\lambda}_{j}(p)|\le c^{1/2}\lVert p\rVert_{\beta}$. By The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension (with $K=H_{\beta}$ and $C=c^{1/2}$) there is a linear map $\widehat{T}:H_{\beta}\to H_{\beta}$ with $\widehat{T}(\widehat{p})=T(p)$ for every $p$ and $|\widehat{T}\zeta|\le c^{1/2}|\zeta|$ for every $\zeta\in H_{\beta}$. Put $\ell(\zeta)=\langle\widehat{T}\zeta,\Omega\rangle_{H_{\beta}}$. Then $\ell$ is linear over $\mathbb{R}$ by bilinearity of the real inner product, and $|\ell(\zeta)|\le|\widehat{T}\zeta|\,|\Omega|\le c^{1/2}|\zeta|$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space; so $\ell$ is a bounded linear functional on $H_{\beta}$, and $\ell(\widehat{p})=\operatorname{Re}\partial^{\lambda}_{j}(p)\,\langle\Omega,\Omega\rangle_{H_{\beta}}=\operatorname{Re}\partial^{\lambda}_{j}(p)$. By The Riesz Representation Theorem for a Real Hilbert Space §existence there is $\xi_{j}\in H_{\beta}=\mathcal{H}_{\lambda}$ with $\ell(\zeta)=\langle\zeta,\xi_{j}\rangle_{H_{\beta}}$ for every $\zeta$.

Let $p\in\mathcal{P}_{d}$. By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §imaginary-unit, $I_{h_{\lambda}}\widehat{p}=I_{h_{\lambda}}J_{\beta}(p)=J_{\beta}(ip)=\widehat{ip}$. By the formula for the pairing in The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion, symmetry of $\langle\cdot,\cdot\rangle_{H_{\beta}}$ and complex-linearity of $\partial^{\lambda}_{j}$,

$$\langle\xi_{j},\widehat{p}\rangle_{\mathcal{H}_{\lambda}}=\langle\xi_{j},\widehat{p}\rangle_{H_{\beta}}-i\,\langle\xi_{j},\widehat{ip}\rangle_{H_{\beta}}=\ell(\widehat{p})-i\,\ell(\widehat{ip})=\operatorname{Re}\partial^{\lambda}_{j}(p)-i\operatorname{Re}\bigl(i\,\partial^{\lambda}_{j}(p)\bigr).$$

Writing $\partial^{\lambda}_{j}(p)=a+bi$ with $a,b$ real (Real and Imaginary Parts of a Complex Number), $i(a+bi)=-b+ai$, so $\operatorname{Re}(i\,\partial^{\lambda}_{j}(p))=-b$ and the right side is $a+bi=\partial^{\lambda}_{j}(p)$. As $j$ was arbitrary, $\lambda$ has conjugate variables $(\xi_{1},\dots,\xi_{d})$ by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate.

Step 4: the bound $\Phi^{*}(\lambda)\le c$. Let $S=\sum_{j}\lVert\xi_{j}\rVert^{2}=\Phi^{*}(\lambda)$ (The Free Fisher Information of a Noncommutative Law §fisher). For each $j\in[d]$ and then each $r\in\mathbb{N}$ choose $q_{j,r}\in\mathcal{P}_{d}$ with $\lVert\widehat{q_{j,r}}-\xi_{j}\rVert<1/r$; this is possible because the classes are dense in $\mathcal{H}_{\lambda}$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense, so the open ball of radius $1/r$ about $\xi_{j}$ meets the set of classes (Dense Subset of a Topological Space). Put $\eta_{j,r}=\widehat{q_{j,r}}$. The real sequence $(1/r)_{r}$ converges to $0$: given $\varepsilon>0$, claim 3 of The Archimedean Property of the Real Numbers gives $N$ with $1/N<\varepsilon$, and $0<1/r\le1/N$ for $r\ge N$.

By the conjugate-variable identity of Step 3 and the Preliminaries, (J) with $p_{j}=q_{j,r}$ reads

$$\Bigl|\sum_{j=1}^{d}\langle\xi_{j},\eta_{j,r}\rangle\Bigr|^{2}\le c\sum_{j=1}^{d}\lVert\eta_{j,r}\rVert^{2}\qquad(r\in\mathbb{N}).$$

By the sesquilinearity recorded in the Preliminaries and claim 1 of The Induced Norm is a Norm, and Induces a Metric, $\bigl|\langle\xi_{j},\eta_{j,r}\rangle-\lVert\xi_{j}\rVert^{2}\bigr|=|\langle\xi_{j},\eta_{j,r}-\xi_{j}\rangle|\le\lVert\xi_{j}\rVert/r$; moreover $\lVert\eta_{j,r}\rVert\le\lVert\xi_{j}\rVert+1$ by the triangle inequality of the norm (claim 2 of The Induced Norm is a Norm, and Induces a Metric), and $\lVert\eta_{j,r}\rVert^{2}-\lVert\xi_{j}\rVert^{2}=\langle\eta_{j,r}-\xi_{j},\eta_{j,r}\rangle+\langle\xi_{j},\eta_{j,r}-\xi_{j}\rangle$ has modulus at most $(2\lVert\xi_{j}\rVert+1)/r$. By claim 6 of Properties of Complex Conjugation and Modulus the real and imaginary parts of these differences are bounded in absolute value by the same quantities, which converge to $0$ by claim 3 of Arithmetic of Limits of Real Sequences; so by claim 3 of Order Properties of Limits of Real Sequences, $\langle\xi_{j},\eta_{j,r}\rangle$ converges to $\lVert\xi_{j}\rVert^{2}$ and $\lVert\eta_{j,r}\rVert^{2}$ converges to $\lVert\xi_{j}\rVert^{2}$ as $r\to\infty$, in the sense of Step 2. As in Step 2, the left side of the last display converges to $S^{2}$ and the right side to $cS$, and claim 1 of Order Properties of Limits of Real Sequences gives $S^{2}\le cS$. If $S>0$, dividing by $S$ gives $S\le c$; if $S=0$, then $S\le c$ since $c\ge0$. Hence $\Phi^{*}(\lambda)\le c$.