(Bodily, 2002) Let g(x) be a stochastic process with zero mean and covariance kernel Cov(g(x), g(y)) = γ (x, y). Let γ (x, y) have sufficient properties so that we can take derivatives g (x) such that...

(Bodily, 2002) Let g(x) be a stochastic process with zero mean and covariance kernel Cov(g(x), g(y)) = γ (x, y). Let γ (x, y) have sufficient properties so that we can take derivatives g (x) such that Cov(g(x), g (y)) = ∂ ∂y γ (x, y) and Cov(g(x), g (y)) = ∂2 ∂x∂y γ (x, y). Consider the computed first difference as having independent random rounding errors e1 and e2 with variance 2 m as

                            h−1 (g(x + h) + e1) − h−1 (g(x) + e2).

 (a) Compute the variance as E  h−1(g(x + h) + e1) − h−1(g(x) + e2) − g(x)2 .

(b) For the covariance kernel γ (x, y) = σ2e−a(x−y)2 , find h (to first order) to minimize this variance.

 (c) Perform a similar analysis for the central difference (f (x + h) − f (x − h))/(2h).