Fractional Brownian motion and optimization by ellipsoids

Let [latex](Z_{t}^{H})_{tin(0,infty)}[/latex] be a fractional Brownian motion with Hurst index H, [latex]delta>0[/latex] and [latex](X_{i}^{H})_{i=1}^{infty}[/latex] be its [latex]delta[/latex] fractional Brownian noise, that is [latex]X_{i}^{H}=Z_{idelta}^{H}-Z_{(i-1)delta}^{H}[/latex]. Set [latex]Y_{i}^{H}=(((X_{2i}^{H}+X_{2i-1}^{H})/(sqrt{2})),((X_{2i}^{H}-X_{2i-1}^{H})/(sqrt2)))[/latex] and let [latex]E_{V}[/latex] be the family of all ellipsoids in the plane centered at the origin with axis at coordinate axis, and with all ellipsoids having the same fixed volume V. Any [latex]Ein E_{V}[/latex] is uniquely determined by its larger axis, say [latex]a[/latex], so [latex]E=E(a)[/latex]. The function [latex]amapsto P(Y_1^{H}in E(a))[/latex] is concave with vanishing derivative at a point explicitly expressible in therms of H. The sequence [latex](Y_{i}^{H})_{i=1}^{infty}[/latex] is stationary and ergodic and therefore [latex](1/n)sum_{i=1}^n1_{E(a)}(Y_{i}^{H})rightarrow E1_{E(a)}[/latex] as [latex]nrightarrowinfty[/latex] almost surely. Since [latex]sum_{i=1}^n1_{E(a)}(Y_{i}^{H})[/latex] counts the number of elements from [latex]{Y_{i}^{H}:i=1,dots,n}[/latex] falling in [latex]E(a)[/latex], optimization of that number respect to [latex]a[/latex], for a given sample from [latex]{Y_{i}^{H}:i=1,dots,n}[/latex], estimates H. A proper choice of V optimizes sensitivity of the method. If [latex](Z_{t}^{H})_{tin(0,infty)}[/latex] is replaced by [latex](sigma Z_{t}^{H}+b)_{tin(0,infty)}[/latex], where [latex]sigma[/latex] and b are unknown constants, a modification of the method coupled with the ergodic method gives an estimation of H, [latex]sigma[/latex], and b. Numerical implementation of the method and its comparison to other existing methods is work in progress.