Most probabilistic models match only some properties of target processes, e.g., current models for wind pressure time series recorded in wind tunnels match the mean and correlation functions [1][2][3][4] or the marginal distributions, in addition to and mean and correlation functions, [5][6][7][8][9][10]. There are no models which match sample properties of target processes, although sample properties are critical for estimating extremes of random processes and related properties [11].
There are at least three reasons for constructing models which capture sample properties, rather than just mean, correlations, and other low order statistics. First, a random process is defined completely by its samples. Mean, variances, correlations, polyspectra and other low order statistics are insufficient to characterize completely random processes, generate samples and estimate extremes of these processes.
Second, processes with the same mean and correlation functions can have very different sample properties and extremes. For example, the processes 𝑋 𝐵 (𝑡) and 𝑋 𝐶 (𝑡) defined by 𝑑𝑋(𝑡) = -𝜌 𝑋(𝑡) 𝑑𝑡 + √ 2 𝜌 𝑑𝑌 (𝑡) with 𝑌 (𝑡) denoting the standard Brownian motion process 𝐵(𝑡) and a compound Poisson process 𝐶(𝑡) have the same mean and correlation functions under proper tuning of the compound Poisson process 𝐶(𝑡). Yet, 𝑋 𝐵 (𝑡) has continuous samples while the samples of 𝑋 𝐶 (𝑡) exhibit jumps at random times. Also, the extremes sup 𝑡∈[0,𝜏] |𝑋 𝐶 (𝑡)| and sup 𝑡∈[0,𝜏] |𝑋 𝐵 (𝑡)| of these processes differ significantly, as illustrated by the histograms of Fig. 1 which are based on 50,000 independent samples of 𝑋 𝐵 (𝑡) and 𝑋 𝐶 (𝑡). Note that the two histograms have different and approximation under the assumption that {𝑍 𝑘 } are independent (solid and dotted lines). a.s., which means that samples of 𝑋 𝑑 (𝑡) can be used as substitutes for samples of 𝑋(𝑡) for sufficiently large 𝑑 so that extremes of 𝑋(𝑡) can be approximated by those of 𝑋 𝑑 (𝑡).
The paper is organized as follows. We define finite dimensional models in Section 2 and give their properties. Conditions under which 𝑋 𝑑 (𝑡) converges weakly to arbitrary processes 𝑋(𝑡) in 𝐶[0, 𝜏] are established in Section 3, which contains our main result. The special cases of Gaussian and translation processes 𝑋(𝑡) are also discussed. Numerical illustrations of our theoretical results are in Section 4. Section 5 summarizes our findings and the Appendix gives computational details for one of the numerical illustrations.
Consider a real-valued process {𝑋(𝑡), 𝑡 ∈ [0, 𝜏]}, 0 < 𝜏 < ∞, defined on a probability space (𝛺, , 𝑃 ) with mean 𝜇(𝑡) = 𝐸[𝑋(𝑡)] = 0 and correlation function 𝑐(𝑠, 𝑡) = 𝐸[𝑋(𝑠) 𝑋(𝑡)]. The assumption 𝜇(𝑡) = 0 is not restrictive since, if 𝜇(𝑡) ≠ 0, the deterministic function 𝜇(𝑡) can be added to the samples of 𝑋(𝑡). It is assumed that the correlation function of 𝑋(𝑡) is continuous, so that it is square integrable on 𝐾 = [0, 𝜏] 2 , i.e., ∫ 𝐾 𝑐(𝑠, 𝑡) 2 𝑑𝑠 𝑑𝑡 < ∞. Under this assumption, 𝜑(𝑡) = ∫ 𝐾 𝑐(𝑠, 𝑡) 𝜑(𝑠) 𝑑𝑠 is a compact, self-adjoint operator on 𝐿 2 (𝐾) so that its eigenvalues {𝜆 𝑘 }, 𝑘 = 1, 2. …, are non-negative and its eigenfunctions {𝜑 𝑘 (𝑡)}, 𝑘 = 1, 2. … are orthonormal, i.e., ⟨𝜑 𝑘 , 𝜑 𝑙 ⟩ = ∫ 𝜏 0 𝜑 𝑘 (𝑡) 𝜑 𝑙 (𝑡) 𝑑𝑡 = 𝛿 𝑘𝑙 . According to Mercer's theorem [14] (Section 6.2) or [15], the series 𝑐(𝑠, 𝑡) = ∑ ∞ 𝑘=1 𝜆 𝑘 𝜑 𝑘 (𝑠) 𝜑 𝑘 (𝑡) converges absolutely and uniformly in 𝐾. Also, 𝑋(𝑡) admits the Karhunen-Loève (KL) representation
where {𝑌 𝑘 } are uncorrelated random variables with 𝐸[𝑌 𝑘 ] = 0 and 𝐸[𝑌 𝑘 𝑌 𝑙 ] = 𝜆 𝑘 𝛿 𝑘𝑙 . The series in (2.1) converges in mean square (m.s.) for any 𝑡 ∈ [0, 𝜏]. This follows from the observation that the FD models
which are truncated versions of 𝑋 KL (𝑡), are such that 𝐸 [( 𝑋 𝐾𝐿,𝑛 (𝑡) -𝑋 𝐾𝐿,𝑚 (𝑡)
) 2 ] = ∑ 𝑛 𝑘=𝑚+1 𝜆 𝑘 𝜑 𝑘 (𝑡) 2 → 0, as 𝑚, 𝑛 → ∞, by Mercer's theorem. This shows that 𝑋 𝐾𝐿,𝑑 (𝑡) is Cauchy in 𝐿 2 [0, 𝜏] and that the series representation of 𝑋 𝐾𝐿 (𝑡) is m.s. convergent [14] (Theorem 6.2.1). Accordingly, 𝑋 KL (𝑡) and 𝑋(𝑡) have the same mean and correlation functions. It can also be shown that 𝐸 [( 𝑋 𝐾𝐿 (𝑠)-𝑋 𝐾𝐿 (𝑡)
The process 𝑋 KL,d (𝑡) in (2.2) is partially specified by its mean and correlation functions which are those of the target process 𝑋(𝑡), unless 𝑋(𝑡) is Gaussian in which case {𝑌 𝑘 } are independent Gaussian variables so that 𝑋 KL,d (𝑡) is a Gaussian process with the first two moments of 𝑋(𝑡). If 𝑋(𝑡) is not Gaussian, the random variables {𝑌 𝑘 } are uncorrelated but dependent non-Gaussian variables. Since the joint distribution of {𝑌 𝑘 } is unknown, it is not possible to generate samples of, e.g., truncated versions 𝑋 𝐾𝐿,𝑑 (𝑡) of 𝑋 KL (𝑡).
We construct an alternative sequence {𝑋 𝑑 (𝑡)} of finite dimensional (FD) processes which is closely related to that in (2.2) in the sense that it shares the same basis functions, i.e., the eigenfunctions {𝜑 𝑘 } of the correlation function of 𝑋(𝑡). It has the expression
where the random coefficients {𝑍 𝑘 } are defined sample-by-sample from samples of 𝑋(𝑡) by projection, i.e.,
where 𝑋(𝑡, 𝜔) denotes a sample of 𝑋(𝑡). We note that (
where the change of order of integration holds by Fubini's theorem. if their components converge in m.s., then their convergence also hold in probability by Chebyshev's inequality. This implies the convergence of the finite dimensional distributions of 𝑋 𝑑 (𝑡) to those of 𝑋(𝑡) as 𝑑 → ∞ [16] (Theorem 18.10).
We note that FD processes of the type in (2.3) can be constructed by using other basis functions, e.g., trigonometric polynomials or other sets of orthogonal functions. We use mainly the eigenfunctions of the correlation functions of 𝑋(𝑡), since they minimize the mean square error and are delivered by available numerical algorithms.
The subsequent section considers processes 𝑋(𝑡) with continuous samples and shows that the sequence of processes {𝑋 𝑑 (𝑡)} converges weakly to 𝑋(𝑡) in the space of continuous functions 𝐶[0, 𝜏] under some conditions. The processes {𝑋 KL,d (𝑡)} do not have this property since their samples are available only for Gaussian target processes 𝑋(𝑡) and, if available, cannot be paired with sample of 𝑋(𝑡) so that the discrepancy between samples of {𝑋 KL,d (𝑡)} and 𝑋(𝑡) cannot be assessed.
We follow the approach of Theorems 8.1 and 8.2 or 8.3 in [17] to show that 𝑋 𝑑 (𝑡) converges weakly to 𝑋(𝑡) in 𝐶[0, 𝜏], a convergence which is denoted by 𝑋 𝑑 (𝑡) 𝑤 → 𝑋(𝑡). Let 𝑋 𝑑 (𝑡), 𝑋(𝑡) ∶ (𝛺, , 𝑃 ) → (𝐶[0, 𝜏], ) be real-valued processes with continuous samples, where denotes the Borel 𝜎-algebra on the space of real-valued continuous functions 𝐶[0, 𝜏]. According to Theorem 8.1, the family of processes {𝑋 𝑑 (𝑡)} converges weakly to 𝑋(𝑡) in 𝐶[0, 𝜏] if (1) the finite dimensional distributions of 𝑋 𝑑 (𝑡) converge to those of 𝑋(𝑡) and (2) the family of processes {𝑋 𝑑 (𝑡)} is tight in 𝐶[0, 𝜏]. We say that the family {𝑋 𝑑 (𝑡)} is tight if for any 𝜀 > 0, there exists a compact set 𝐾 ⊂ 𝐶[0, 𝜏] such that 𝑃 (𝑋 𝑑 (𝑡) ∈ 𝐾) > 1 -𝜀 for all 𝑑. Theorems 8.2 and 8.3 provide criteria for checking whether a sequence of probability measures is tight, and we use the conditions of these theorems to determine whether the family {𝑋 𝑑 (𝑡)} of processes is tight. Since we already have shown the convergence of the finite dimensional distributions of 𝑋 𝑑 (𝑡) to those of 𝑋(𝑡), we only need to show the tightness of the family {𝑋 𝑑 (𝑡)} of processes to prove that 𝑋 𝑑 (𝑡) converges weakly to 𝑋(𝑡) in 𝐶[0, 𝜏]. The following theorem is our main result.
where 𝑋 𝑑 (𝑡) is given by (2.3).
(ii) There is 𝑀 > 0 such that 𝐸[ sup 𝑡∈[0,𝜏] | Ẋ𝑑 (𝑡)| ] ≤ 𝑀 for all 𝑑 ≥ 1 and {𝜑 𝑘 (𝑡)}, 𝑘 ≥ 1, are continuously differentiable functions.
Proof. Following Theorem 8.2 in [17], we first show the tightness of the family of random variables {𝑋 𝑑 (0)}. Note that
by Mercer's theorem and 𝐸𝑋(0) 2 is finite by assumption. Then for any 𝜀 > 0 there exists 𝐿 > 0 such that (𝑡). We show that the sequence {𝑋 𝑑 (𝑡)} of processes satisfies this condition provided that (𝑖) or (𝑖𝑖) holds.
Case one (𝑖) holds: for given 𝜀 > 0, we have
Since 𝐿 𝑘 (𝛿) is monotonically increasing, ∑ ∞ 𝑘=1 𝐸|𝑍 𝑘 |𝐿 𝑘 (𝜏) < ∞ and each 𝜑 𝑘 (𝑡) is continuous so that 𝐿 𝑘 (𝛿) is bounded, then by dominated convergence theorem [18] (Theorem 1.34), we know that for any 𝜀, 𝜂 > 0, there exists 𝛿 0 such that 𝑃 ( sup
where 𝐹 is the joint distribution function of (𝑍
since 𝜑 𝑘 (𝑡) is assumed to be continuously differentiable, which implies
Note that 𝜉 is between 𝑠 and 𝑡 and 𝑡 ∈ [𝑠 -𝛿, 𝑠 + 𝛿], then
Hence 𝑋 𝑑 (𝑡) is tight in 𝐶[0, 𝜏]. This property and the convergence of the finite dimensional distributions of 𝑋 𝑑 (𝑡) to those of 𝑋(𝑡) imply
In applications, we can use the condition (𝑖) or (𝑖𝑖) depending on properties of 𝑋(𝑡). For example, the condition (𝑖𝑖) holds for m.s. differentiable processes, see Theorem 3.
by properties of the Gaussian variables and the fact that ∑ 𝑑 𝑘=1 𝜆 𝑘 𝜓 𝑘 (𝑠, 𝑡) 2 increases with 𝑑 and is bounded. This shows that the second condition of the Theorem 12.3 in [17] is satisfied for 𝛾 = 4, 𝛼 = 2 and the monotonically increasing and continuous function ℎ(𝑡) = √ 3𝑡, so that 𝐵 𝑑 converges weakly to 𝐵 in 𝐶 [0,1]. This also implies the convergence
It is not surprising that the Brownian motion process 𝐵(𝑡) does not satisfy the conditions of our main result although the family {𝐵 𝑑 (𝑡)} of its FD models converges weakly to 𝐵(𝑡) in 𝐶[0, 1] since the statement of the theorem does not make any assumption on the distribution of the target process 𝑋(𝑡). In contrast, the above proof of the remark uses explicitly the fact that 𝐵(𝑡) is a Gaussian process with independent increments.
We now develop conditions for the weak convergence of FD models {𝑋 𝑑 (𝑡)} for Gaussian processes 𝑋(𝑡) with smooth samples based on the condition (ii) of Theorem 3.1.
sup
Proof. Following Theorem 8.3 in [17], we first show that the sequence {𝐺 𝑑 (0)} of random variables is tight. This follows from the observation that for any 𝜀 > 0, there exists 𝐿 such that
Then, we show that the sequence of processes
which implies
where 2 ] is as small as desired. Then for any 𝜀, 𝜂 > 0, there exists 𝛿 0 such that sup 𝑡∈[𝑠,𝑠+𝛿 0 ] 𝐸[(𝐺(𝑡) -𝐺(𝑠)) 2 ] ≤ 𝜀 2 ∕(4 log(2𝐾∕𝜂)). Further, for any 𝑑 ≥ 1, we have
Therefore the second condition of Theorem 8.3 in [17] holds, which means that sup
is the sum of independent normal random variables for fixed 𝑡, i.e., the random variables 𝑍 𝑘 𝜑 𝑘 (𝑡), then 𝐺 𝑑 converges a.s to 𝐺 in 𝐶[0, 𝜏] by Itô-Nisio theorem [19]. □ We extend the above result to a class of non-Gaussian processes, referred to as translation processes, which are monotonically increasing mappings of Gaussian processes. Let 𝑋(𝑡) be a translation process defined by
where 𝐺(𝑡) is a stationary Gaussian process with zero mean and unit variance, 𝛷 denotes the distribution of the standard normal variable and 𝐹 is the marginal distribution of 𝑋(𝑡). The translation processes {𝑋(𝑡)} are completely defined by the marginal distribution 𝐹 and the correlation function of 𝐺(𝑡). Translation processes exist if the selected marginal distributions and the correlation functions satisfy some compatibility conditions [22]. Generally, these conditions are mild, since the correlation functions of the translation processes and their Gaussian images are similar.
Since 𝐹 -1 is continuous, then 𝐹 -1 is uniformly continuous on [0, 1], which leads to sup 𝑡∈[0,𝜏]
As previously stated, there are other FD models in addition to those in (2.3). The following theorem considers FD models 𝑋 (𝑁) (𝑡) whose samples interpolate linearly between values of 𝑋(𝑡) at the times (0, 𝛥𝑡, … , 𝑁 𝛥𝑡), where 𝛥𝑡 = 𝜏∕𝑁. The samples of these FD models are continuous functions so that they are elements of 𝐶[0, 𝜏]. Under the conditions of the following theorem, the discrepancy between the samples of 𝑋 (𝑁) (𝑡) and those of 𝑋(𝑡) measured by the metric of 𝐶[0, 𝜏] can be made as small as desired by increasing 𝑁. Generally, the stochastic dimension 𝑁 + 1 of 𝑋 (𝑁) (𝑡) is much larger than that of 𝑋 𝑑 (𝑡) in (2.3) so that they are less useful in applications. (3.4)
Proof. As previously, we show that the finite dimensional distributions of 𝑋 (𝑁) (𝑡) converge to those of 𝑋(𝑡), the sequence of random variables {𝑋 (𝑁) (0)} is tight and the sequence of processes
This implies (𝑋 (𝑁) (𝑡 1 ), … , 𝑋 (𝑁) (𝑡 𝑛 ))
𝑚.𝑠.
→ (𝑋(𝑡 1 ), … , 𝑋(𝑡 𝑛 )) for any 𝑛 ≥ 1 and 𝑡 1 , … , 𝑡 𝑛 ∈ [0, 𝜏] which extends to convergence in probability by Chebyshev's inequality. The latter yields the convergence of the finite dimensional distributions of 𝑋 (𝑁) (𝑡) to those of 𝑋(𝑡) as 𝑁 → ∞ by Theorem 18.10 in [16].
Note that the sequence {𝑋 (𝑁) (0)} of random variables is tight, since for any 𝜀 > 0, there exists 𝐿 such that
We now show the tightness of 𝑋 (𝑁)
) .
( sup
) , The following corollaries describe several special models that converge weakly in the continuous space 𝐶[0, 𝜏] under some conditions. We define the finite dimensional models 𝑋 𝑑 (𝑡) and apply Theorem 3.1 to determine whether sup 𝑡∈[0,𝜏] |𝑋 𝑑 (𝑡) -𝑋(𝑡)| 𝑤 → 0 or not. Corollary 3.1. Let 𝑔(𝜈), 𝜈 ≥ 0, denote the one-sided spectral density of a zero-mean weakly stationary process 𝑋(𝑡), 0 ≤ 𝑡 ≤ 𝜏. Consider the sequence of processes {𝑋 𝑑 (𝑡)}, 0 ≤ 𝑡 ≤ 𝜏, which are obtained from 𝑋(𝑡) by truncating its spectral density to 𝑔 𝑑 (𝜈) = 𝑔(𝜈) 1(𝜈 ≤ 𝜈 𝑑 ), where
[ sup
which takes the form
where 𝑐 𝑘 ≥ 0, 𝑐 𝑘 = 𝑐 -𝑘 , 𝜈 1 = 2 𝜋∕𝑇 and 𝜈 𝑘 = 𝑘 𝜈 1 , 𝜈 𝑘 = -𝜈 -𝑘 . The series ∑ ∞ 𝑘=1 𝑐 𝑘 is convergent since 𝑋(𝑡) has finite variance by assumption. Processes with the second moment properties in (3.6) are referred to as mean square periodic. The spectral representation of 𝑋(𝑡) has the form [24] (Section 3.9.4)
where [25] (Section 1.10). These series have the form
where 𝜈 1 = 2 𝜋∕𝑇 * , 𝜈 𝑘 = 𝑘 𝜈 1 and
(3.12)
We define 𝑋 * 𝑑 (𝑡) by truncating the spectral representation of 𝑋 * (𝑡),
Proof. We cannot apply Corollary 3.2, because the models in (3.7) and (3.10) differ, e.g., 𝑋 * (𝑡) is not stationary. We first show that 𝑋 * 𝑑 (𝑡) converges to 𝑋 * (𝑡) in mean square sense. For any 𝑚 > 𝑛, we have
This shows that {𝑋 * 𝑑 (𝑡)} is Cauchy, since
→ 𝑋 * (𝑡), which implies the convergence of the finite dimensional distributions of 𝑋 * 𝑑 (𝑡) to those of 𝑋 * (𝑡). Further, by Theorem 3.1 the sequence of processes {𝑋 * 𝑑 (𝑡)} has the property sup 𝑡∈
We illustrate numerically that samples and extremes of real-valued continuous processes 𝑋(𝑡) can be approximated by samples of FD models {𝑋 𝑑 (𝑡)}, 𝑑 = 1, 2, …, of these processes for sufficiently large 𝑑 provided that they satisfy the conditions of Theorem 3.1 and related results. The discrepancy between samples of 𝑋(𝑡) and 𝑋 𝑑 (𝑡) is measured by the metric sup 0≤𝑡≤𝜏 |𝑋(𝑡, 𝜔) -𝑋 𝑑 (𝑡, 𝜔)| of the space 𝐶[0, 𝜏] of continuous samples. This discrepancy cannot be obtained exactly since the samples of these processes can only be recorded at finite numbers of times, e.g., the times {𝑡 𝑘 }, 𝑘 = 0, 1, … , 𝑁, where 𝑡 𝑘 = 𝑡 𝑘-1 + 𝛥𝑡 and 𝛥𝑡 = 𝜏∕𝑁. The numerical value, max 𝑘=0,1,…,𝑁 |𝑋(𝑡 𝑘 , 𝜔) -𝑋 𝑑 (𝑡 𝑘 , 𝜔)|, of the discrepancy between 𝑋(𝑡) and 𝑋 𝑑 (𝑡) provides a lower bound on sup 0≤𝑡≤𝜏 |𝑋(𝑡, 𝜔) -𝑋 𝑑 (𝑡, 𝜔)|. Since 𝑋(𝑡) and 𝑋 𝑑 (𝑡) have continuous samples, it is expected that the lower bound is tight for sufficiently large 𝑁.
Accordingly, we approximate the samples of 𝑋(𝑡) by those of random vectors ( 𝑋(𝑡 0 ), 𝑋(𝑡 1 ) … , 𝑋(𝑡 𝑁 ) )
, and the actual discrepancy sup 0≤𝑡≤𝜏 |𝑋(𝑡, 𝜔) -𝑋 𝑑 (𝑡, 𝜔)| between samples of 𝑋(𝑡) and 𝑋 𝑑 (𝑡) by its numerical values max 𝑘=0,1,…,𝑁 |𝑋(𝑡 𝑘 , 𝜔) -𝑋 𝑑 (𝑡 𝑘 , 𝜔)| for large 𝑁. In some of examples, we only prove the weak convergence of 𝑋 𝑑 to 𝑋 in 𝐶[0, 𝜏]. Under this convergence, the measure of the set
can be made as small as desired for any 𝜀 > 0 by increasing 𝑑. Generally, the sets 𝛺 𝑑 (𝜀) differ for 𝑑 ≠ 𝑑 ′ but have small measures for sufficiently large 𝑑 and 𝑑 ′ . Therefore, although we may not have almost sure convergence, we can still use the samples of 𝑋 𝑑 (𝑡) as substitutes for samples of 𝑋(𝑡) for sufficiently large 𝑑, since the probability that samples of 𝑋 𝑑 (𝑡) may misrepresent samples of 𝑋(𝑡) is 𝑃 (𝛺 𝑑 (𝜀)).
This section illustrates the construction of FD models 𝑋 𝑑 (𝑡) for three real-valued processes 𝑋(𝑡), a stationary Gaussian process, a nonstationary Gaussian process and a non-Gaussian translation process, and (2) quantifies the performance of the resulting FD models 𝑋 𝑑 (𝑡) by two metrics, the norm sup 0≤𝑡≤𝜏 |𝑋(𝑡) -𝑋 𝑑 (𝑡)| and the discrepancy between extremes sup 0≤𝑡≤𝜏 |𝑋(𝑡)| and sup 0≤𝑡≤𝜏 |𝑋 𝑑 (𝑡)| of 𝑋(𝑡) and 𝑋 𝑑 (𝑡). The examples show that under the conditions of our theoretical results, samples and extremes of 𝑋 𝑑 (𝑡) can be used as substitutes for samples and extremes of 𝑋(𝑡) provided that the stochastic dimension 𝑑 is sufficiently large.
Example 4.1. Let 𝑋(𝑡), 0 ≤ 𝑡 ≤ 𝜏, be a real-valued process defined by the differential equation
with the initial conditions 𝑋(0) = 0 and Ẋ(0) = 0, where 𝛼, 𝛽 > 0 are constants, 𝑌 is the stationary solution of 𝑑𝑌 (𝑡) = -𝜌 𝑌 (𝑡) 𝑑𝑡 + √ 2 𝜌 𝑑𝐵(𝑡), 𝜌 > 0, and 𝐵 denotes the standard Brownian motion.
The mean and correlation functions of 𝑌 (𝑡) are 𝐸[𝑌 (𝑡)] = 0 and 𝐸[𝑌 (𝑠) 𝑌 (𝑡)] = exp(-𝜌 |𝑠 -𝑡|) for any 𝜌 > 0. Since the Brownian motion has continuous samples, the processes 𝑋(𝑡), Ẋ(𝑡) and 𝑌 (𝑡) also have continuous samples as they are obtained from samples of 𝐵(𝑡) by integration. These processes are Gaussian as linear transformations of 𝐵(𝑡). The target process is the solution 𝑋(𝑡) of (4.1) which is a non-stationary Gaussian process with continuous samples.
The basis functions {𝜑 𝑘 (𝑡)} for the FD models 𝑋 𝑑 (𝑡) of 𝑋(𝑡) in (4.1) are the top eigenfunctions of the correlation function 𝑐(𝑠, 𝑡) = 𝐸[𝑋(𝑠) 𝑋(𝑡)] of 𝑋(𝑡), i.e., the eigenfunctions corresponding to the largest 𝑑 eigenvalues of 𝑐(𝑠, 𝑡), 0 ≤ 𝑠, 𝑡 ≤ 𝜏. The correlation function of 𝑋(𝑡) can be obtained by solving the differential equations for the correlation function of the vector-valued process ( 𝑋(𝑡), Ẋ(𝑡), 𝑌 (𝑡) ) , see [24], Section 7.2.1.1, or by estimation from samples of 𝑋(𝑡). The basis functions {𝜑 𝑘 (𝑡)} were obtained by (A.1) in Appendix. FD models of 𝑋(𝑡) can also be obtained from FD models 𝐵 𝑑 (𝑡) of the Brownian motion 𝐵(𝑡) and the defining equation of 𝑋(𝑡) but this approach was not followed.
We first show that samples of 𝑋(𝑡) can be approximated by samples of 𝑋 𝑑 (𝑡) for sufficiently large 𝑑, then present numerical estimates for the discrepancy between samples of 𝑋(𝑡) and 𝑋 𝑑 (𝑡). The mean square (m.s.) convergence 𝑋 𝑑 (𝑡) m.s.
→ 𝑋(𝑡) for a fixed time 𝑡 follows from Mercer's theorem. This also implies the m.s. convergence of ( 𝑋 𝑑 (𝑠 1 ), … , 𝑋 𝑑 (𝑠 𝑚 ) ) to ( 𝑋(𝑠 1 ), … , 𝑋(𝑠 𝑚 ) ) as 𝑑 → ∞ for arbitrary 0 ≤ 𝑠 1 < ⋯ < 𝑠 𝑚 ≤ 𝜏 and 𝑚 ≥ 1, then their convergence also hold in probability by Chebyshev's inequality, which leads to the convergence of the finite dimensional distributions of 𝑋 𝑑 (𝑡) to those of 𝑋(𝑡), see [16], Theorem 18.10. Since the process 𝑋(𝑡) is Gaussian with zero mean and finite variance, we also have the a.s. convergence of 𝑋 𝑑 to 𝑋 in 𝐶[0, 𝜏], see Theorem 3.2. Accordingly, we can substitute samples of 𝑋(𝑡) with samples of 𝑋 𝑑 (𝑡) for sufficiently large 𝑑.
The ∑ ∞ 𝑘=1 𝑐 𝑘 = 1. The sequence of processes 𝑋 𝑑 (𝑡) has the form in (3.9) so that
→ 0 by Theorem 3.2 since 𝑋 𝑑 (𝑡) and 𝑋(𝑡) are Gaussian processes. Hence, samples of 𝑋(𝑡) can be substituted with samples of 𝑋 𝑑 (𝑡) for sufficiently large 𝑑.
As previously stated, since it is not possible to generate the samples of target process 𝑋(𝑡), we approximate this process by 𝑋 n(𝑡), which is set equal to 𝑋 𝑑 (𝑡) in (3.9