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A light curve is the brightness of an astronomical source as a function of time, expressed as flux or apparent magnitude. A periodic light curve repeats after a characteristic time; Gaussian process models can reconstruct irregularly sampled light curves and propagate observational uncertainty.

A mathematical model represents selected features of a system with mathematical objects and relations so that its behavior can be analyzed or predicted.

A Gaussian process has a mean-square derivative at $t$ when its difference quotients converge in L2 space. The mean-square convergence need not give differentiable sample paths. For a centered stationary process with twice continuously differentiable covariance function $K(t-s)$, the difference quotients are Cauchy in L2 space, since their covariance is a rectangular average of $-K^{\prime \prime }$. Their limits form a Gaussian process with covariance function $\mathbb{E}[Y(s)Y(t)]=-K^{\prime \prime }(s-t)$.

If an L2 space-valued map $X$ has a continuous mean-square derivative $Y$, then its increments equal the Bochner integral of $Y$. To see this, for each partition interval $[t_j,t_{j+1}]$, continuous linear observations and the scalar mean-value bound give $\Vert X(t_{j+1})-X(t_j)-(t_{j+1}-t_j)Y(t_j){\Vert }_2\le (t_{j+1}-t_j){\sup }_{u\in [t_j,t_{j+1}]}\Vert Y(u)-Y(t_j){\Vert }_2$. Sum these errors. Uniform continuity of $Y$ makes the total error tend to zero as the mesh tends to zero, while the Riemann sums converge to the Bochner integral. This proves the fundamental theorem of calculus in L2 space. If a Gaussian process derivative also has a continuous modification, its anchored path integral $X(s)+{\int }_s^t\tilde{Y}(u)du$ therefore produces a differentiable modification of a stochastic process of the original Gaussian process.

Let $W_n$ be the interpolation in part (a). The operationis continuous, since $|I(f)-I(g)|\le \Vert f-g{\Vert }_{\infty }$. By the Donsker invariance principle and the continuous mapping theorem,The exact trapezoidal integral of the linear interpolation isTherefore the statistic in question differs from $I(W_n)$ by $S_n/(2n^{3/2})$. Using independence, zero means, and unit variances givesThis error tends to zero in $L^2$, hence in probability. The Slutsky theorem now proves the integrated random-walk limit:The limiting law can also be made explicit. The time integral is a Gaussian random variable, as a mean-square limit of linear combinations of a Gaussian process. It is centered, and the covariance identity $\mathbb{E}(B_s{B}_t)=\min (s,t)$ givesThus the terminal value of integrated Brownian motion here has law $N(0,1/3)$.

Choose independent random variables ${\xi }_i\sim N(0,1)$ on a countable product probability space. For each fixed $t$, the diagonal assumption givesThe partial sums ${\sum }_{i\le n}{k}_i(t){\xi }_i$ are therefore Cauchy in $L^2$. DefineChoose a representative of this limit for each $t$. No path continuity or simultaneous series convergence over all uncountably many times is being asserted.

For any finite list $t_1,\dots ,t_m$ and real coefficients $a_j$, the linear combination ${\sum }_j{a}_j{X}_{t_j}$ is the $L^2$ limit of centered Gaussian variablesTheir variances converge, so their characteristic functions converge to that of a centered normal distribution. This proves that every finite-dimensional vector is Gaussian and hence that $X$ is a Gaussian process. Taking $L^2$ limits also givesThis is the Gaussian process construction from square-summable features.