Rice's Formula for the Expected Number of Upcrossings

Derives Lemma A22.3. Joint Gaussianity. Fix \(a,b\in\R\) and write \(D_{h}=\left(X(m+h)-X(m)\right)/h\), a linear combination of two values of \(X\); by (H1) the variable \(S_{h}=aX(m)+bD_{h}\) is Gaussian with mean zero, say \(S_{h}\sim\mathcal{N}(0,v_{h})\). Put \(S=aX(m)+bX'(m)\). By (H3), \(D_{h}\rightarrow X'(m)\) in mean square, so \(S_{h}\rightarrow S\) in mean square and hence in probability, and therefore in distribution. Mean-square convergence also forces \(v_{h}=\avg{S_{h}^{2}}\rightarrow\avg{S^{2}}=v\), since \(\abs{\avg{S_{h}^{2}}^{1/2}-\avg{S^{2}}^{1/2}} \le\avg{(S_{h}-S)^{2}}^{1/2}\) by the triangle inequality for the second-moment norm. The characteristic function of \(S_{h}\) is \(\ee^{-v_{h}t^{2}/2}\) (Proposition 15.29 and the computation of the Gaussian transform in the derivation of Theorem 15.46), which converges to \(\ee^{-vt^{2}/2}\) for every \(t\). By Corollary A18.14 the characteristic function of \(S\) is that limit, and by Corollary A18.11 \(S\) is therefore \(\mathcal{N}(0,v)\). As \(a\) and \(b\) were arbitrary, the pair is jointly Gaussian.

The covariance vanishes. By the Cauchy–Schwarz inequality (Lemma 15.63, applied without centring, both variables having mean zero), \(\abs{\avg{X(m)\left(D_{h}-X'(m)\right)}} \le\avg{X(m)^{2}}^{1/2}\avg{\left(D_{h}-X'(m)\right)^{2}}^{1/2} \rightarrow0\), so

where \(\pp_{2}\) is the derivative in the second slot, which exists and is continuous by (H2). Now \(r(m,m)=1\) for every \(m\) by (H1); differentiating this identity in \(m\) by the chain rule (Proposition 11.104) gives \(\pp_{1}r(m,m)+\pp_{2}r(m,m)=0\), while \(r\) is symmetric, \(r(n,n')=r(n',n)\), so \(\pp_{1}r(m,m)=\pp_{2}r(m,m)\). Hence both vanish, and \(\avg{X(m)X'(m)}=0\) by Equation (A22.3). Together with \(\operatorname{var}X(m)=r(m,m)=1\) and \(\operatorname{var}X'(m)=\sigma_{1}(m)^{2}>0\) from (H4), the covariance matrix is \(\operatorname{diag}(1,\sigma_{1}^{2})\), which is invertible.

Independence. A jointly Gaussian pair with mean zero and invertible covariance \(\Sigma\) has density \(\left(2\pi\right)^{-1}\left(\det\Sigma\right)^{-1/2} \exp\left(-\tfrac12\vect{y}\transpose\Sigma^{-1}\vect{y}\right)\); with \(\Sigma\) diagonal the quadratic form splits and the density factorises as Equation (A22.2), which by Definition 15.25 is independence.

Derives Lemma A22.4. Finiteness. Let \(Z=\set{m\in M\mid g(m)=z}\), a closed subset of the compact \(M\) and hence compact; by hypothesis it misses both endpoints. If \(p\in Z\) then \(g'(p)\neq0\), and \(g'\) is continuous, so \(g\) is strictly monotone on a neighbourhood of \(p\) and takes the value \(z\) there only at \(p\): every point of \(Z\) is isolated. A compact set all of whose points are isolated is finite — the singletons form an open cover with no proper subcover. Write \(Z=\set{p_{1}<\dots<p_{J}}\), all interior to \(M\).

A separating radius. For each \(j\) choose \(\eta>0\), the same for all \(j\), small enough that the closed intervals \(U_{j}=[p_{j}-\eta,p_{j}+\eta]\) are pairwise disjoint, contained in \((m_{1},m_{2})\), and such that \(g'\) keeps the sign of \(g'(p_{j})\) throughout \(U_{j}\); continuity of \(g'\) and finiteness of \(Z\) make this possible. On \(M\setminus\bigcup_{j}\left(p_{j}-\eta,p_{j}+\eta\right)\) — a compact set on which \(g-z\) does not vanish — the continuous function \(\abs{g-z}\) attains a strictly positive minimum \(\mu\) (Theorem 11.24). Let \(\delta_{0}=\min\left(\mu,\;\min_{j}\abs{g(p_{j}-\eta)-z},\; \min_{j}\abs{g(p_{j}+\eta)-z}\right)\), which is positive because \(g\) is strictly monotone on each \(U_{j}\) with \(g(p_{j})=z\).

The integral, for \(\delta<\delta_{0}\). The set \(A_{\delta}=\set{m\in M\mid\abs{g(m)-z}<\delta}\) is contained in \(\bigcup_{j}U_{j}\), since \(\delta<\mu\). Fix \(j\). On \(U_{j}\) the function \(g\) is strictly monotone and continuous, so \(A_{\delta}\cap U_{j}\) is the open interval \(\left(\alpha_{j},\beta_{j}\right)\) whose endpoints are the two solutions of \(g=z\mp\delta\) inside \(U_{j}\); these exist and lie in the interior of \(U_{j}\) by the intermediate value theorem (Theorem 11.23) and the choice \(\delta<\delta_{0}\).

If \(g'>0\) on \(U_{j}\) — an upcrossing at \(p_{j}\), by Definition 15.100, since \(g\) passes from below \(z\) to above it there — then \(\left(g'\right)^{+}=g'\) on \(U_{j}\) and, by the second fundamental theorem of calculus (Theorem 11.43),

If instead \(g'<0\) on \(U_{j}\), then \(\left(g'\right)^{+}=0\) there and the contribution is \(0\); that is a downcrossing and is correctly not counted. Summing Equation (A22.5) over the finitely many cells, the integral in Equation (A22.4) equals \(2\delta\) times the number of \(j\) with \(g'(p_{j})>0\), which is \(N_{z}(g)\); dividing by \(2\delta\) gives the identity, with a value independent of \(\delta\).

Derives Lemma A22.8. By Equation (15.20) and the substitution \(w=\sigma y\) (Equation (11.27)),

Derives Lemma A22.9. By (H1) the variable \(X(m)\) is standard Gaussian for every \(m\), so \(\Pr(\abs{X(m)-z}<\delta)=\int_{z-\delta}^{z+\delta}\varphi(v)\dd v\), independently of \(m\). Since \(\varphi\) is continuous, the mean value theorem for integrals — apply Theorem 11.35 to the antiderivative \(\Phi\), whose derivative is \(\varphi\) by Proposition 15.22 — gives \(\int_{z-\delta}^{z+\delta}\varphi=2\delta\,\varphi(\zeta_{\delta})\) for some \(\zeta_{\delta}\) in the interval. Applying Theorem 11.35 once more, this time to \(\varphi\) itself, \(\abs{\varphi(\zeta_{\delta})-\varphi(z)}\le\delta\sup_{v} \abs{\varphi'(v)}\), and \(\varphi'(v)=-v\varphi(v)\) has modulus maximised at \(v=\pm1\), where it equals \(\ee^{-1/2}/\sqrt{2\pi}\).

Proof of Theorem A22.2. Derives Theorem A22.2. By (H3) and Remark A22.5, almost every sample path satisfies the hypotheses of Lemma A22.4, so almost surely

Take expectations. The dominated-convergence clause of Theorem A22.6 gives \(\avg{N_{z}(X)}=\lim_{\delta\downarrow0}\avg{Y_{\delta}}\), and the Tonelli clause — the integrand is non-negative — gives, for each fixed \(\delta\),

At each fixed \(m\) the indicator is a function of \(X(m)\) alone and the positive part a function of \(X'(m)\) alone, and those two are independent by Lemma A22.3; so the expectation factorises, and by Lemmas A22.8 and A22.9,

From Equations (A22.2) and (A22.9) (factorise the expectation at each \(m\) by independence, then evaluate the level factor and the mean positive part). The point \(\zeta_{\delta}\) does not depend on \(m\) (Lemma A22.9), so it comes outside the integral:

the integral being finite because \(\sigma_{1}\) is continuous on the compact \(M\) (H4) and therefore bounded and Riemann integrable (Theorem 11.40). Letting \(\delta\downarrow0\) and using the uniform estimate of Equation (A22.7),

since \(\varphi(z)/\sqrt{2\pi}=\ee^{-z^{2}/2}/(2\pi)\). That is Equation (A22.1).

Derives Corollary A22.10. With \(D_{h}\) as in Lemma A22.3, the second-moment norm is continuous under mean-square convergence, so \(\avg{D_{h}D_{h'}}\rightarrow\avg{X'(m)^{2}}\) as \(h,h'\rightarrow0\). On the other hand

a second-order difference quotient of the twice continuously differentiable \(r\), which converges to \(\pp_{1}\pp_{2}r(m,m)\). Hence \(\sigma_{1}(m)^{2}=\pp_{1}\pp_{2}r(m,m)\) in general. Now write \(r(n,n')=\varrho(n-n')\) with \(\varrho\) twice continuously differentiable by (H2) and \(\varrho(0)=1\); then \(\pp_{1}r=\varrho'(n-n')\) and \(\pp_{2}\pp_{1}r=-\varrho''(n-n')\), so \(\sigma_{1}(m)^{2}=-\varrho''(0)=\lambda_{2}\), independent of \(m\), and the integral in Equation (A22.1) is \(L\sqrt{\lambda_{2}}\).