#  M-brane singularity formation

J.Eggers1 and J.Hoppe2 1School of Mathematics, University of Bristol,
University Walk, Bristol BS8 1TW, United Kingdom
2Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm,
Sweden

###### Abstract

We derive self-similar string solutions in a graph representation, near the
point of singularity formation, which can be shown to extend to point-like
singularities on M-branes, as well as to the radially symmetric case.

##  I Introduction

For more than 40 years, Barbashov and Chernikov (1966, 1967, 1968) (see also
Whitham (1974)), various ways of solving the non-linear equation

$\ddot{z}(1+z^{\prime
2})-z^{\prime\prime}(1-\dot{z}^{2})=2\dot{z}z^{\prime}\dot{z}^{\prime}$ (1)

are known. Recent work on higher dimensional time-like zero mean curvature
hypersurfaces includes Christodoulou (2007); Milbredt (2008); Hoppe (2008);
Bellettini et al. (2008) Here we show that (1) can develop singularities in
finite time, starting from finite initial data. The structure of these
singularities is described by the self-similar ansatz

$z(t,x)=z_{0}-\hat{t}+\hat{t}^{\alpha}h\left(\frac{x}{\hat{t}^{\beta}}\right)+\dots$
(2)

where $\hat{t}:=t_{0}-t\rightarrow 0$ (the dots are indicating lower order
terms). Inserting (2) into (1) one finds the above ansatz to be consistent
provided $\beta=(1+\alpha)/2>1$, and

$h^{\prime\prime}\left(2\alpha
h-\frac{(\alpha+1)^{2}}{4}\xi^{2}\right)=(\alpha-1)\left[h^{\prime 2}+\alpha
h-\frac{3}{4}(\alpha+1)\xi h^{\prime}\right]\;,$ (3)

for (2) to be an asymptotic solution of (1). For consistency with a finite
outer solution of (1), the profile $h$ must satisfy

$h(\xi)\propto
A_{\pm}\xi^{\frac{2\alpha}{\alpha+1}}\quad\mbox{for}\quad\xi\rightarrow\pm\infty$
(4)

(for a general discussion of matching self-similar solutions to the exterior
see Eggers and Fontelos (2008)).

The ansatz (2) is formally consistent for a continuum of similarity exponents
$\alpha\geq 1$ and for any solution of the similarity equation (3). However,
by considering the regularity of solutions of (3) in the origin $\xi=0$ the
similarity exponent must be one of the sequence

$\alpha=\alpha_{n}=\frac{n+1}{n},\quad n\in\mathbb{N},$ (5)

certainly if $h(0)=0=h^{\prime}(0)$, and presumably in general (i.e. all
relevant solutions of (3)). Of this infinite sequence, we believe that only
$\alpha=2$ is realized for generic initial data; indeed, in this case

$\xi=\zeta+c\zeta^{3}/3,\quad h(\xi)=\zeta^{2}/2+c\zeta^{4}/4,$ (6)

which we will deduce from a parametric string solution corresponding to (1).

The importance of the similarity solution (2) lies in the fact that it can be
generalized to arbitrary dimensions, in particular to membranes. We find that
the same type of singular solution is observed in any dimension, even having
the same spatial structure (6).

##  II The similarity equation

A way of satisfying (3) is to demand

$L^{2}:=h^{\prime 2}+2\alpha h-(\alpha+1)\xi h^{\prime}=0.$ (7)

(differentiating e.g. $(1+\alpha)\xi=h^{\prime}+2\alpha h/h^{\prime}$ one can
eliminate $h^{\prime\prime}$, reducing (3) to an identity, as long as
$h^{\prime}\neq 1$).

The transformation

$h(\xi)=\xi^{2}g(\xi)=\xi^{2}\left(\frac{(1+\alpha)^{2}}{8\alpha}-\frac{v^{2}}{2\alpha}\right)$
(8)

yields

$-\frac{d\xi}{\xi}=\frac{vdv}{v^{2}\pm\alpha
v+(\alpha^{2}-1)/4}=\frac{1}{2}\left(\frac{\alpha+1}{v\mp\frac{\alpha+1}{2}}-\frac{\alpha-1}{v\mp\frac{\alpha-1}{2}}\right)dv,$
(9)

i.e. (choosing the upper sign)

$\frac{\left|v-(\alpha+1)/2\right|^{\alpha+1}}{\left|v-(\alpha-1)/2\right|^{\alpha-1}}=\frac{E}{\xi^{2}}.$
(10)

This yields solutions $v\in[(\alpha-1)/2,(\alpha+1)/2)$,

$\displaystyle
v\approx\frac{\alpha-1}{2}+\left(\frac{\xi^{2}}{E}\right)^{\frac{1}{\alpha-1}}+\dots\quad\mbox{as}\quad\xi\rightarrow
0$ (11) $\displaystyle
v\approx\frac{\alpha+1}{2}-\left(\frac{\xi^{2}}{E}\right)^{\frac{1}{\alpha+1}}+\dots\quad\mbox{as}\quad\xi\rightarrow\pm\infty,$

i.e.

$\displaystyle h(\xi)\geq 0,h(0)=0,$ (12) $\displaystyle
h(\xi)\propto\xi^{2}/2\quad\mbox{as}\quad\xi\rightarrow 0$ $\displaystyle
h(\xi)\propto\frac{1+\alpha}{2\alpha}\xi^{\frac{2\alpha}{1+\alpha}}\quad\mbox{as}\quad\xi\rightarrow\pm\infty.$

Note that these solutions are consistent with the growth conditions (4).

To solve the second order equation (3) we note that
$\tilde{h}(\xi):=c\;h(\xi/\sqrt{c})$ solves (3), if $h$ does, and that

$\frac{h^{\prime}}{\xi}-\frac{2h}{\xi^{2}}=\frac{1}{\alpha}f\left(\sqrt{\frac{(\alpha+1)^{2}}{4}-2\alpha\frac{h(\xi)}{\xi^{2}}}\right)\quad\equiv\left(\frac{1}{\alpha}f(v)\right)$
(13)

reduces (3) to

$-\left(v^{2}-\frac{(\alpha+1)^{2}}{4}\right)\left(v^{2}-\frac{(\alpha-1)^{2}}{4}\right)=f\left(\alpha
vf^{\prime}-(\alpha-1)f-(\alpha+2)v^{2}+(\alpha^{2}-1)(\alpha-2)/4\right),$
(14)

and

$\frac{d\xi}{\xi}=-\frac{vdv}{f(v)}=\frac{\alpha dg}{f}.$ (15)

The growth condition (4) implies that $h$ grows less than quadratically at
infinity. Thus we deduce from (13) that $f$ vanishes at $(\alpha+1)/2$.
Furthermore, from a direct calculation using the growth exponent (4) we find
the first derivative, yielding the initial conditions

$f\left(\frac{\alpha+1}{2}\right)=0,\quad
f^{\prime}\left(\frac{\alpha+1}{2}\right)=1.$ (16)

Using (16), (14) yields a polynomial solution

$f(v)=\left(v-\frac{(\alpha+1)}{2}\right)\left(v-\frac{(\alpha-1)}{2}\right)=v^{2}-\alpha
v+\frac{\alpha^{2}-1}{4},$ (17)

(i.e.) (10), which corresponds to the first order equation (7), but also an
infinity of other solutions (a Taylor expansion around
$v_{\infty}=(\alpha+1)/2$ shows that (14) leaves
$f^{\prime\prime}((\alpha+1)/2)$ undetermined, when (16) holds). We note that
(14) also has the solution $f_{-}(v)=f(-v)$, and for the special case
$\alpha=2$ another pair of polynomial solutions,

$\tilde{f}(v)=\left(v+3/2\right)\left(v-1/2\right)=v^{2}+v-3/4,$ (18)

and $\tilde{f}_{-}(v)=\tilde{f}(-v)$. While the asymptotic behavior following
from (18) is in disagreement with (4), integration methods similar to those
used by Abel Abel (1881) perhaps permit a complete reduction of (14) to
quadratures.

In any case, (14) can be simplified in various ways. For $\alpha=2$, e.g. it
reduces to

$yy^{\prime}=y-\frac{1}{4v^{5/2}}(v^{2}-9/4)(v^{2}-1/4)$ (19)

via

$f(v)=\sqrt{v}y(4v^{3/2}/3).$ (20)

The solution (17), which is consistent with the growth condition (4), is
equivalent to the solution (10) of (7) given before. If one investigates the
behavior of the solution in the origin (either using (10) directly or by
series expansion of (7)), one finds that only for $\alpha=\alpha_{n}$ (cf.
(5)) a smooth solution is possible. Thus the first consistent solution is
found for $n=1$ or $\alpha=2$. Higher order solutions $n=2,3,\dots$ are also
possible in principle. They have the property that apart from
$f^{\prime\prime}(0)$, the first non-vanishing derivative is $f^{(2n+2)}(0)$.
However, we believe that they correspond to non-generic initial conditions,
whose derivatives have corresponding properties of vanishing up to a certain
order. To demonstrate this point, one would have to perform a stability
analysis of the corresponding solution Eggers and Fontelos (2008). In the
string picture discussed below this can be shown explicitly, as higher order
solutions correspond to non-generic initial data.

##  III Higher dimensions

The solutions of (7), found to govern singularities of (1), also apply to
higher dimensions. The reason is that the left hand side of (7) is the leading
order term of

${\cal L}^{2}=1-\dot{z}^{2}+z^{\prime 2}.$ (21)

In other words, the asymptotic singular solutions discussed above have ${\cal
L}^{2}=0+$ lower order. In fact, differentiating (21) with respect to $t$ and
$x$ one easily shows that ${\cal L}^{2}=0$ provides solutions of (1). In
higher dimensions, differentiating $1-z^{\alpha}z_{\alpha}=0$ gives
$z^{\alpha}z_{\alpha\beta}=0$, and hence

$(1-z_{\alpha}z^{\alpha})\square z+z^{\beta}z^{\alpha}z_{\alpha\beta}=0.$ (22)

Thus solutions of ${\cal L}^{2}=0$ also solve the M-brane equation (22) in
arbitrary dimensions.

For the special case of radially symmetric membranes:

$\ddot{z}(1+z^{\prime
2})-z^{\prime\prime}(1-\dot{z}^{2})-2\dot{z}z^{\prime}\dot{z}^{\prime}=\frac{z^{\prime}}{r}\left(1-\dot{z}^{2}+z^{\prime
2}\right)\equiv\frac{z^{\prime}}{r}{\cal L}^{2}.$ (23)

Insert the radial version of the ansatz (2),

$z(t,r)=-\hat{t}+\hat{t}^{\alpha}h\left(\frac{r-r_{0}}{\hat{t}^{\beta}}\right)+\dots,$
(24)

into (23). If $r_{0}\neq 0$, the entire right hand side of (23) is of lower
order in $\hat{t}$, and the similarity equation (3) remains the same.
Geometrically, this corresponds to the singularity forming along a circular
ridge of radius $r_{0}$.

If on the other hand $r_{0}=0$, i.e. the singularity forms along the axis, the
right hand side is of the same order, and the similarity equation becomes

$h^{\prime\prime}\left(2\alpha
h-\frac{(\alpha+1)^{2}}{4}\xi^{2}\right)+(1-\alpha)\left[h^{\prime 2}+\alpha
h-\frac{3}{4}(\alpha+1)\xi
h^{\prime}\right]=-\frac{h^{\prime}}{\xi}\left[h^{\prime 2}+2\alpha
h-(\alpha+1)\xi h^{\prime}\right].$ (25)

This equation can in principle have solutions different from (3). For
solutions of (7), however, the expression in angular brackets in (25)
vanishes, hence solutions of (7) also solve (25). Thus (24),(6) describe a
point-like singularity on a membrane. These observations straightforwardly
generalize to higher M-branes, $M>2$.

##  IV Parametric string solution

Let us now compare our findings with the solution of closed bosonic string
motions given by equation (50) of Hoppe (1995). (note that the definitions of
$f$ and $g$ are changed by $\pi/4$, and that the constant $\lambda$ is chosen
to be 1):

$\dot{{\bf x}}(t,\varphi)=\sin(f-g)\left(\begin{array}[]{c}-\sin(f+g)\\\
\cos(f+g)\end{array}\right)$ (26) ${\bf
x}^{\prime}(t,\varphi)=\cos(f-g)\left(\begin{array}[]{c}\cos(f+g)\\\
\sin(f+g)\end{array}\right),$ (27)

where $f=f(\varphi+t)$ and $g=g(\varphi-t)$. From (27) one finds the curvature

$k(t,\varphi)=\frac{f^{\prime}+g^{\prime}}{\cos(f-g)}.$ (28)

The hodograph transformation

$\displaystyle(t,\varphi)\rightarrow t=x^{0},\quad x=x^{1}(t,\varphi),$ (29)
$\displaystyle x^{2}(t,\varphi)=z(t,x^{1}(t,\varphi)),$

implying $\dot{z}=\dot{y}-\dot{x}y^{\prime}/x^{\prime}$,
$z^{\prime}=y^{\prime}/x^{\prime}$, $(\partial\phi/\partial
x^{0}=-\dot{x}/x^{\prime},(\partial\phi/\partial x^{1}=1/x^{\prime})$ permits
to go between the parametric string picture (26)-(28) and the graph
description (1). In particular,

$1-\dot{z}^{2}+z^{\prime 2}=\left(\frac{\cos(f-g)}{\cos(f+g)}\right)^{2}$ (30)

is manifestly non-negative in the parametric string-description, while for
solutions of (1) one has to demand it explicitly - leading e.g. to the
exclusion of solutions with $h^{\prime}(0)=0$, $h(0)<0$.

Let us give an explicit example of $\mathbb{M}_{2}\subset\mathbb{R}^{1,2}$
being at $t=0$ a regular graph, while for $t=1$ a curvature singularity has
developed. Let $\mathbb{M}_{2}$ be described by ${\bf x}(\varphi,t)$, as
defined by (26),(27), with $\varphi\in\mathbb{R}$, $t\geq 0$. Let

$\displaystyle f(w)=\left\\{\begin{array}[]{ll}\arctan w&\mbox{for}\quad
w\geq\epsilon\\\ \chi_{\epsilon}(w)\arctan w&\mbox{for}\quad 0\leq
w\leq\epsilon<0\\\ 0&\mbox{for}\quad w\leq 0\end{array}\right.,$ (34)

where $\chi_{\epsilon}(w\geq\epsilon)=1$, $\chi_{\epsilon}(w\leq 0)=0$, and
$\chi_{\epsilon}(0<w<0)$ such that $f^{\prime}(w)\geq 0$. We also assume that
$g(w)=-f(-w)$. A simple calculation then shows that for
$\varphi\in[-t+\epsilon,t-\epsilon]$ one obtains (${\bf x}_{0}(t=0,u=0)=0$)

$\displaystyle x(\varphi,t)=-\varphi+\arctan(\varphi+t)+\arctan(\varphi-t)$
(35) $\displaystyle
y(\varphi,t)=\ln\sqrt{\left(1+(\varphi+t)^{2}\right)\left(1+(\varphi-t)^{2}\right)}$
$\displaystyle
k(\varphi,t)=\frac{2}{\sqrt{\left(1+(\varphi+t)^{2}\right)\left(1+(\varphi-t)^{2}\right)}}\frac{\varphi^{2}+1+t^{2}}{\varphi^{2}+1-t^{2}}$

Note that for $t>1$ this is no longer a graph.

Figure 1:  The formation of a swallowtail, as described by (40). Shown is a
smooth minimum, ($\hat{t}>0$), a minimum with a 4/3 singularity ($\hat{t}=0$),
and a swallowtail or double cusp ($\hat{t}<0$).

The example (34) underlies a general structure that can be uncovered by a
local expansion of the functions $f$ and $g$ around the singularity. Namely,
as seen from (28), the singularity occurs when $f-g$ is a multiple of $\pi/2$;
for simplicity, we also assume that $g(w)=-f(-w)$ as before.

Then a local Taylor expansion yields

$f(\zeta)=\pi/4+a(\zeta-\zeta_{0})-b(\zeta-\zeta_{0})^{2}+O(\zeta-\zeta_{0})^{3},$
(36)

so together with the symmetry requirement we find

$f-g=\pi/2+2a(t-\zeta_{0})-2b\varphi^{2}+(t-\zeta_{0})^{2},$ (37)

where we neglected higher-order terms in the expansion, which will turn out to
be irrelevant for the singularity formation.

From this expression it is clear that $\zeta_{0}$ has to be identified with
the singular time $t_{0}$ and $a>0$ for the solution to be regular for
$t<t_{0}$. Similarly, one must have $b>0$ (otherwise $f-g$ would be $\pi/2$ at
an earlier time), and the singularity occurs at $\varphi=0$. Thus to leading
order we have

$f-g\approx\pi/2-2a\hat{t}-2b\varphi^{2},\quad f+g\approx 2a\varphi,$ (38)

from which we get

$x^{\prime}=2a\hat{t}+2b\varphi^{2},\quad
y^{\prime}=4a^{2}\hat{t}\varphi+4ab\varphi^{3}.$ (39)

Integrating this expression, using the integrability condition (27), gives

$x=\hat{t}\varphi+c\varphi^{3}/3,\quad
y=-\hat{t}+\hat{t}\varphi^{2}/2+c\varphi^{4}/4,$ (40)

where we have used a rescaling of the parameter $\varphi$.

The curve described by (40) is shown in Fig. 1, but disregarding the spatial
translation of $z$ by the term $-\hat{t}$. In catastrophe theory, this is
known as the swallowtail Arnold (1984). For $\hat{t}>0$ the curve is smooth,
while for $\hat{t}=0$ a rather mild singularity develops; at the origin,
$y\propto x^{4/3}$. After the singularity ($\hat{t}<0$) the curve self-
intersects. The solution (40) leads directly to the similarity form (2), if
one notices that $\varphi$ is of order $\hat{t}^{1/2}$ for the terms in (40)
to balance. Thus, using the notation of (2), and putting
$\zeta=\varphi/\hat{t}^{1/2}$, one finds $\alpha=2,\beta=3/2$ for the
exponents, and (6) for the similarity function. The crucial point is that
although (6) came out of an expansion, all higher order terms are subdominant
in the limit $\hat{t}\rightarrow 0$. Thus (6) is in fact an exact solution of
(3) with $\alpha=2$. Moreover, it is even a solution of (7), and precisely of
the form (10).

In the case of non-generic initial conditions other solutions are possible.
For example, instead of (36)

$f(\zeta)=\pi/4+a(\zeta-\zeta_{0})-b(\zeta-\zeta_{0})^{2n},$ (41)

where $n\in\mathbb{N}$ but $n>1$. Only even powers $2n$ are allowed, otherwise
the singularity occurs for all $\varphi$ at the same time, i.e. it is no
longer point-like. If the leading order term is not linear but itself of
higher order, the resulting similarity profile becomes singular at the origin,
cf. 51. Repeating the above calculation along the same lines, we find

$x^{\prime}=2a\hat{t}+2b\varphi^{2n},\quad
y^{\prime}=4a^{2}\hat{t}\varphi+4ab\varphi^{2n+1},$ (42)

which is equivalent to the symmetric shape function

$\displaystyle\xi=\zeta+2d(n+1)\zeta^{2n+1}/(2n+1)$ (43) $\displaystyle
h=\zeta^{2}/2+d\zeta^{2n+2}.$

The corresponding similarity exponent is $\alpha=\alpha_{n}$, as given by (5).
We thus retrieve the exact same solutions identified by our previous analysis,
based on a similarity description.

*

##  Appendix A Additional solutions of the similarity equation

It is possible to construct many more solutions to the similarity equation
(7), which are all defined on the real line, but which we reject since they
either contradict (4) or are not smooth. The simplest case is

$h(\xi)=\frac{\xi^{2}}{2},$ (44)

which is a solution for any $\alpha$, but evidently does not satisfy the
matching condition (4).

Recall that for $\alpha=\alpha_{n}$, (10) furnishes smooth solutions on the
real line. On the other hand, while for $\alpha=3$ the second derivative of
the resulting solution is well-defined, the third derivative is discontinuous.
Namely, for $E=4$ (e.g.) one finds that for $\xi>0$,

$\displaystyle h(\xi)=-\frac{2}{3}\left(\xi+2(1+\xi)-2(1+\xi)^{3/2}\right)$
(45) $\displaystyle h^{\prime}(\xi)=2\left(\sqrt{1+\xi}-1\right)>0$
$\displaystyle h^{\prime\prime}(\xi)=1/\sqrt{1+\xi},$

so that

$\displaystyle
h^{\prime\prime\prime}(\xi)=\left\\{\begin{array}[]{ll}-(1+\xi)^{-3/2}/2&\mbox{for}\quad\xi>0\\\
(1+\xi)^{-3/2}/2&\mbox{for}\quad\xi<0.\\\ \end{array}\right.$ (48)

Other solutions whose scaling exponent is not from the set (5), but which have
well-defined second derivatives, can be found from the parametric string
solution as described in section IV. If the expansion of $f$ does not start
with a linear term as in (36), but at higher order, e.g.

$f(\zeta)=\pi/4+(\zeta-\zeta_{0})^{3}/2-b(\zeta-\zeta_{0})^{4},$ (49)

one finds instead of (39)

$x^{\prime}=3\hat{t}\varphi^{2}+2b\varphi^{4},\quad
y^{\prime}=3\hat{t}\varphi^{5}+2b\varphi^{7}.$ (50)

Integrating (50), the result once more conforms with (2), with a similarity
exponent of $\alpha=4$, and the similarity function has the parametric form

$\displaystyle\xi=\zeta^{3}+2b\zeta^{5}/5$ (51) $\displaystyle
h=\zeta^{6}/2+b\zeta^{8}/4.$

It is confirmed easily that (51) solves (7) with $\alpha=4$, but the third
derivative of $h(\xi)$ is singular at the origin.

###### Acknowledgements.

J.H. would like to thank P.T. Allen and J. Isenberg for a discussion and
correspondence, as well as the Swedish Research Council and the Marie Curie
Training Network ENIGMA (contract MRNT-CT-2004-5652) for financial support.

## References

  * Barbashov and Chernikov (1966) B. M. Barbashov and N. A. Chernikov, Sov. Phys. JETP 23, 861 (1966).
  * Barbashov and Chernikov (1967) B. M. Barbashov and N. A. Chernikov, Sov. Phys. JETP 24, 437 (1967).
  * Barbashov and Chernikov (1968) B. M. Barbashov and N. A. Chernikov, Sov. Phys. JETP 27, 971 (1968).
  * Whitham (1974) G. B. Whitham, _Linear and Nonlinear Waves_ (John Wiley & Sons, 1974).
  * Christodoulou (2007) D. Christodoulou, _The Formation of Shocks in 3-dimensional Fluids_ (EMS Monographs in Mathematics, 2007).
  * Milbredt (2008) O. Milbredt, _The Cauchy problem for membranes_ , Ph.D. Thesis, FU Berlin (2008).
  * Hoppe (2008) J. Hoppe, _Non-linear realization of Poincaré invariance in the graph-representation of extremal hypersurfaces_ (2008), URL http://arxiv:0806.0656.
  * Bellettini et al. (2008) G. Bellettini, M. Novaga, and G. Orlandi, _Time-like Lorentzian minimal submanifolds as singular limits of nonlinear wave equations_ (2008).
  * Eggers and Fontelos (2008) J. Eggers and M. A. Fontelos, _The role of self-similarity in singularities of partial differential equations_ (2008), URL http://arXiv:0812.1339.
  * Abel (1881) N. H. Abel, _Oeuvres II_ (Imprimerie de Grøndahl & Søn, Christiania, 1881).
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  * Arnold (1984) V. I. Arnold, _Catastrophe Theory_ (Springer, 1984).


$\bm{w}^{\top}B_{2,i}\,\bm{w}=\bm{w}^{\top}(A_{2,i}+A_{2,i}^{\top})\,\bm{w}=2\,\bm{w}^{\top}A_{2,i}\,\bm{w}$.

Assuming A2.6, we find that

$\sum_{i,j\in\mathcal{F}}\bm{w}^{\top}D_{i,j}\,\bm{w}\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}=\widehat{\bm{w}}^{\top}D\,\widehat{\bm{w}}\geq
0,$

where
$\widehat{\bm{w}}=(\bm{w}\,\tau_{i}^{\beta_{i}/2})_{i\in\mathcal{F}}\in\mathbb{R}^{\mathcal{F}}$.
Therefore,

(82)
$\sum_{i,j=1}^{n}\Xi_{i,j}\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}=2\sum_{i=1}^{n}\bm{w}^{\top}A_{2,i}\,\bm{w}\,\tau_{i}^{\beta_{i}}+\underbracket{\sum_{i,i\in\mathcal{F}}\bm{w}^{\top}D_{i,j}\,\bm{w}\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}}_{\geq\,0}\geq
2\sum_{i=1}^{n}\bm{w}^{\top}A_{2,i}\,\bm{w}\,\tau_{i}^{\beta_{i}}.$

####  8.3.3. Generalized variance expansion

######  Lemma 18.

The generalized variance

$\sigma_{\bm{b}}^{-2}(\bm{\tau})=\min_{\bm{x}\geq\bm{b}}\bm{x}^{\top}\,\Sigma^{-1}(\bm{\tau})\,\bm{x}$

admits the following asymptotic formula:

(83)
$\sigma_{\bm{b}}^{-2}(\bm{\tau})-\sigma_{\bm{b}}^{-2}(\bm{0})=\sum_{i,j=1}^{n}\Xi_{i,j}\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}+o\left(\sum_{i=1}^{n}\tau_{i}^{\beta_{i}}\right),$

where $\Xi$ is defined by (80) or (81).

###### Proof of Lemma 18.

We remind the reader of our notation convention
$A_{I}=A_{II}=(A_{ij})_{i,\,j\in I}$ for a given $d\times d$ matrix $A$ and
$I\subset\\{1,\ldots,d\\}$.

Define a vector-valued function $\bm{b}(\bm{\tau})$ by

$\sigma_{\bm{b}}^{-2}(\bm{\tau})=\min_{\bm{x}\geq\bm{b}}\bm{x}^{\top}\,\Sigma^{-1}(\bm{\tau})\,\bm{x}\eqqcolon\bm{b}^{\top}(\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\,\bm{b}(\bm{\tau}).$

We have

(84)
$\sigma_{\bm{b}}^{-2}(\bm{\tau})-\sigma_{\bm{b}}^{-2}(\bm{0})=\bm{b}^{\top}(\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\,\bm{b}(\bm{\tau})-\bm{b}^{\top}(\bm{0})\,\Sigma^{-1}(\bm{0})\,\bm{b}(\bm{0}).$

Adding and subtracting $\bm{b}(\bm{0})$ and using Lemma 5, we obtain

$\displaystyle\bm{b}^{\top}(\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\,\bm{b}(\bm{\tau})$
$\displaystyle=\bm{b}^{\top}(\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\,\bm{b}(\bm{0})+\bm{b}(\bm{\tau})\,\Sigma(\bm{\tau})\,\Big{[}\bm{b}(\bm{\tau})-\bm{b}(\bm{0})\Big{]}$
$\displaystyle=\bm{b}^{\top}(\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\,\bm{b}(\bm{0})+\bm{b}_{I(\bm{\tau})}^{\top}(\bm{\tau})\,\Sigma_{I(\bm{\tau})}^{-1}(\bm{\tau})\,\Big{[}\bm{b}(\bm{\tau})-\bm{b}(\bm{0})\Big{]}_{I(\bm{\tau})}.$

Take $\varepsilon(\bm{0})>0$ as in Lemma 7 and let
$|\bm{\tau}|<\varepsilon(\bm{0})$. Then, we have that $I(\bm{\tau})\subset
I(\bm{0})$, and therefore
$\bm{b}_{I(\bm{\tau})}(\bm{\tau})=\bm{b}_{I(\bm{\tau})}=\bm{b}_{I(\bm{\tau})}(\bm{0})$,
hence the last term on the right is zero. Similarly, adding and subtracting
$\bm{b}(\bm{\tau})$ in the second term of (84) and applying Lemma 5, we obtain

$\bm{b}^{\top}(\bm{0})\,\Sigma^{-1}(\bm{0})\,\bm{b}(\bm{0})=\bm{b}^{\top}(\bm{\tau})\,\Sigma^{-1}(\bm{0})\,\bm{b}(\bm{0})+\Big{[}\bm{b}(\bm{0})-\bm{b}(\bm{\tau})\Big{]}_{I(\bm{0})}^{\top}\,\Sigma_{I(\bm{0})}^{-1}(\bm{0})\,\bm{b}_{I(\bm{0})}(\bm{0}).$

The last term is zero, since $I(\bm{0})\subset I(\bm{t})\cup K(\bm{t})$, and
therefore
$\bm{b}_{I(\bm{0})}(\bm{0})=\bm{b}_{I(\bm{0})}=\bm{b}_{I(\bm{0})}(\bm{\tau})$.
Therefore,

(85)
$\sigma_{\bm{b}}^{-2}(\bm{\tau})-\sigma_{\bm{b}}^{-2}(\bm{0})=\bm{b}^{\top}(\bm{\tau})\,\Big{[}\Sigma^{-1}(\bm{\tau})-\Sigma^{-1}(\bm{0})\Big{]}\,\bm{b}(\bm{0}).$

Note the small difference with the formula (79): here we have
$\bm{b}(\bm{\tau})$ on the left instead of $\bm{b}=\bm{b}(\bm{0})$. This,
however, does not matter within the given error, since by A2.3 and (73) the
difference

$\Big{[}\bm{b}(\bm{\tau})-\bm{b}(\bm{0})\Big{]}^{\top}\Big{[}\Sigma^{-1}(\bm{\tau})-\Sigma^{-1}\Big{]}\,\bm{b}(\bm{0})=o\left(\sum_{i,j=1}^{n}\tau_{i}^{\beta_{i}^{\prime}}\,\tau_{j}^{\beta_{j}^{\prime}}\right)$

is within the required error. By (78), we arrive at (83). ∎

####  8.3.4. Conditional mean

######  Lemma 19.

The conditional mean vector

$\bm{d}_{\,\bm{\tau}}(\bm{t})=\Big{[}I-R(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\Big{]}\bm{b}$

admits the following asymptotic formula

(86)
$\bm{d}_{\,\bm{\tau}}(\bm{t})=\sum_{i=1}^{n}\Bigg{[}A_{2,i}\Big{[}(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\Big{]}+S_{\alpha_{i},A_{5,i}}(t_{i})\Bigg{]}\bm{w}\\\\[7.0pt]
+\sum_{i,j\in\mathcal{F}}\tau_{j}^{\beta_{j}/2}\,\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}D_{i,j}\,\bm{w}+\epsilon(\bm{\tau},\bm{t}),$

with $D_{i,j}=A_{6,i,j}+A_{1,i}\Sigma^{-1}A_{1,j}^{\top}$ and the error
$\epsilon$ satisfying

$\epsilon(\bm{\tau},\bm{t})=o\left(\sum_{i=1}^{n}\Big{[}\tau_{i}^{\beta_{i}}+t_{i}^{\beta_{i}}+|t_{i}|^{\alpha_{i}}\Big{]}\right).$

###### Proof of Lemma 19.

Throughout this calculation, the equivalence relation $f\sim g$ is defined as
follows:

$f\sim g\iff
f-g=o\left(\sum_{i=1}^{n}\Big{[}t_{i}^{\beta_{i}}+\tau_{i}^{\beta_{i}}+|t_{i}|^{\alpha_{i}}\Big{]}\right).$

In order to simplify the computation, define $C$ and $B$ by

$R(\bm{t},\bm{s})\eqqcolon\Sigma-C,\qquad\Sigma^{-1}(\bm{\tau})\eqqcolon\Sigma^{-1}+\Sigma^{-1}\,B\,\Sigma^{-1}.$

With these shorthands, we have

$I-R(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}(\bm{\tau})=I-\Big{[}\Sigma-C\Big{]}\Big{[}\Sigma^{-1}+\Sigma^{-1}B\,\Sigma^{-1}\Big{]}=(C-B)\,\Sigma^{-1}+C\,\Sigma^{-1}B\,\Sigma^{-1}$

Note that

$\displaystyle C$
$\displaystyle\sim\sum_{i=1}^{n}\Big{[}A_{1,i}\,(\tau_{i}+t_{i})^{\beta_{i}^{\prime}}+A_{2,i}\,(\tau_{i}+t_{i})^{\beta_{i}}+A_{1,i}^{\top}\,\tau_{i}^{\beta_{i}^{\prime}}+A_{2,i}^{\top}\,\tau_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(t_{i})\Big{]}+\sum_{i,\,j\in\mathcal{F}}A_{6,i,j}\,(\tau_{i}+t_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},$
$\displaystyle B$
$\displaystyle\sim\sum_{i=1}^{n}\Big{[}B_{1,i}\,\tau_{i}^{\beta_{i}^{\prime}}+B_{2,i}\,\tau_{i}^{\beta_{i}}\Big{]}+\sum_{i,j\in\mathcal{F}}(A_{6,i,j}+B_{1,i}\Sigma^{-1}B_{1,j})\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},$

with $B_{k,i}=A_{k,i}+A_{k,i}^{\top}$, $k=1,2$. First term:

$(C-B)\,\Sigma^{-1}\sim\sum_{i=1}^{n}\Bigg{[}A_{1,i}\left[(\tau_{i}+t_{i})^{\beta_{i}^{\prime}}-\tau_{i}^{\beta_{i}^{\prime}}\right]+A_{2,i}\left[(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\right]+S_{\alpha_{i},A_{5,i}}(t_{i})\Bigg{]}\Sigma^{-1}\\\\[7.0pt]
-\sum_{i,j\in\mathcal{F}}B_{1,i}\Sigma^{-1}B_{1,j}\Sigma^{-1}\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}+\sum_{i,j\in\mathcal{F}}A_{6,i,j}\Sigma^{-1}\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}\,\tau_{j}^{\beta_{j}/2}.$

Second term:

$C\,\Sigma^{-1}B\,\Sigma^{-1}\sim\sum_{i,j\in\mathcal{F}}\Big{[}A_{1,i}\Sigma^{-1}B_{1,j}(\tau_{i}+t_{i})^{\beta_{i}/2}\tau_{j}^{\beta_{j}/2}+A_{1,i}^{\top}\Sigma^{-1}B_{1,j}\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}\Big{]}\Sigma^{-1}.$

Adding the two formulae together, we observe a few cancellations:

(87)
$I-R(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\sim\sum_{i=1}^{n}\Bigg{[}A_{1,i}\Big{[}(\tau_{i}+t_{i})^{\beta_{i}^{\prime}}-\tau_{i}^{\beta_{i}^{\prime}}\Big{]}+A_{2,i}\Big{[}(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\Big{]}+S_{\alpha_{i},A_{5,i}}(t_{i})\Bigg{]}\Sigma^{-1}\\\\[7.0pt]
+\sum_{i,j\in\mathcal{F}}\tau_{j}^{\beta_{j}/2}\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}\Big{[}A_{6,i,j}+A_{1,i}\Sigma^{-1}B_{1,j}\Big{]}\Sigma^{-1}.$

Multiplying by $\bm{b}$ on the right and using $A_{1,i}\,\bm{w}=\bm{0}$ yields
(86). ∎

####  8.3.5. Conditional covariance

######  Lemma 20.

The conditional covariance function

$\mathcal{R}_{\bm{\tau}}(\bm{t},\bm{s})\coloneqq
R(\bm{\tau}+\bm{t},\bm{\tau}+\bm{s})-R(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\,R(\bm{\tau},\bm{\tau}+\bm{s})$

admits the following asymptotic formula

(88)
$\mathcal{R}_{\bm{\tau}}(\bm{t},\bm{s})=\sum_{i=1}^{n}\Big{[}S_{\alpha_{i},A_{5,i}}(t_{i})+S_{\alpha_{i},A_{5,i}}(-s_{i})-S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})\Big{]}\\\\[7.0pt]
+\sum_{i,j\in\mathcal{F}}D_{i,j}\left((\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\right)\left((\tau_{j}+s_{j})^{\beta_{j}/2}-\tau_{j}^{\beta_{j}/2}\right)+\epsilon(\bm{\tau},\bm{t},\bm{s})$

with $D_{i,j}=A_{6,i,j}+A_{1,i}\Sigma^{-1}A_{1,j}^{\top}$ and the error
$\epsilon$ satisfying

(89)
$\epsilon(\bm{\tau},\bm{t},\bm{s})=o\left(\sum_{i=1}^{n}\Big{[}t_{i}^{\beta_{i}}+s_{i}^{\beta_{i}}+\tau_{i}^{\beta_{i}}+|t_{i}|^{\alpha_{i}}+|s_{i}|^{\alpha_{i}}+|t_{i}-s_{i}|^{\alpha_{i}}\Big{]}\right)$

as $(\bm{\tau},\bm{t},\bm{s})\to\bm{0}$.

###### Proof of Lemma 20.

Let us, for the sake of clarity, denote

(90)
$R(\bm{t},\bm{s})\eqqcolon\Sigma-C(\bm{t},\bm{s}),\qquad\Sigma^{-1}(\bm{\tau})\eqqcolon\Sigma^{-1}+\Sigma^{-1}\,B(\bm{\tau})\,\Sigma^{-1}.$

Therefore, we have

$\displaystyle\mathcal{R}(\bm{\tau},\bm{t},\bm{s})$
$\displaystyle=\Sigma-C(\bm{\tau}+\bm{t},\bm{\tau}+\bm{s})$
$\displaystyle-\Big{[}\Sigma-C(\bm{\tau}+\bm{t},\bm{\tau})\Big{]}\Big{[}\Sigma^{-1}+\Sigma^{-1}\,B(\bm{\tau})\,\Sigma^{-1}\Big{]}\Big{[}\Sigma-C(\bm{\tau},\bm{\tau}+\bm{s})\Big{]}$
$\displaystyle=-C(\bm{\tau}+\bm{t},\bm{\tau}+\bm{s})+C(\bm{\tau}+\bm{t},\bm{\tau})+C(\bm{\tau},\bm{\tau}+\bm{s})-B(\bm{\tau})+F+\epsilon,$

where

$F\coloneqq
B(\bm{\tau})\,\Sigma^{-1}\,C(\bm{\tau},\bm{\tau}+\bm{s})+C(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}\,B(\bm{\tau})-C(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}\,C(\bm{\tau},\bm{\tau}+\bm{s})$

and

(91)
$\epsilon\coloneqq-C(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}\,B(\bm{\tau})\,\Sigma^{-1}\,C(\bm{\tau},\bm{\tau}+\bm{s})=O\left(\sum_{i=1}^{n}\left[\tau_{i}^{3\beta_{i}^{\prime}}+t_{i}^{3\beta_{i}^{\prime}}+s_{i}^{3\beta_{k}^{\prime}}+|t_{i}|^{3\alpha_{i}}+|s_{i}|^{3\alpha_{i}}\right]\right)$

can be subsumed into (89). Let us substitute $C$ and $B$ with their
expressions

(92) $\displaystyle C(\bm{t},\bm{s})$
$\displaystyle\sim\sum_{i=1}^{n}\Big{[}A_{1,i}\,t_{i}^{\beta_{i}^{\prime}}+A_{2,i}\,t_{i}^{\beta_{i}}+A_{1,i}^{\top}\,s_{i}^{\beta_{i}^{\prime}}+A_{2,i}^{\top}\,s_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})\Big{]}+\sum_{i,j\in\mathcal{F}}A_{6,i,j}\,t_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2},$
$\displaystyle B(\bm{\tau})$
$\displaystyle\sim\sum_{i=1}^{n}\Big{[}B_{1,i}\,\tau_{i}^{\beta_{i}^{\prime}}+B_{2,i}\,\tau_{i}^{\beta_{i}}\Big{]}+\sum_{i,j\in\mathcal{F}}\Big{[}A_{6,i,j}+B_{1,i}\Sigma^{-1}B_{1,j}\Big{]}\,\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}.$

Let us first compute the coefficients in front of various powers of $\tau$,
$\tau+t$ and $\tau+s$ in

(93)
$-C(\bm{\tau}+\bm{t},\bm{\tau}+\bm{s})+C(\bm{\tau}+\bm{t},\bm{\tau})+C(\bm{\tau},\bm{\tau}+\bm{s})-B(\bm{\tau}).$

Coefficients of the following terms are zero:

$(\tau_{i}+t_{i})^{\beta_{i}^{\prime}},\ (\tau_{i}+t_{i})^{\beta_{i}},\
(\tau_{i}+s_{i})^{\beta_{i}^{\prime}},\ (\tau_{i}+s_{i})^{\beta_{i}},\
\tau_{i}^{\beta_{i}^{\prime}},\ \tau_{i}^{\beta_{i}}.$

Next, we proceed to the mixed terms:

$\displaystyle(\tau_{i}+t_{i})^{\beta_{i}/2}\,(\tau_{j}+s_{j})^{\beta_{j}/2}\colon\quad$
$\displaystyle-A_{6,i,j},\qquad$
$\displaystyle\tau_{i}^{\beta_{i}/2}\,(\tau_{j}+s_{j})^{\beta_{j}/2}\colon\quad$
$\displaystyle A_{6,i,j},$
$\displaystyle(\tau_{i}+t_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}\colon\quad$
$\displaystyle A_{6,i,j},\qquad$
$\displaystyle\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}\colon\quad$
$\displaystyle-A_{6,i,j}-B_{1,i}\Sigma^{-1}B_{1,j}$

Let us compare these with the corresponding orders in

$F=B(\bm{\tau})\,\Sigma^{-1}\,C(\bm{\tau},\bm{\tau}+\bm{s})+C(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}\,B(\bm{\tau})-C(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}\,C(\bm{\tau},\bm{\tau}+\bm{s}).$

We have:

$\displaystyle(\tau_{i}+t_{i})^{\beta_{i}/2}\,(\tau_{j}+s_{j})^{\beta_{j}/2}\colon$
$\displaystyle-A_{1,i}\Sigma^{-1}A_{1,j}^{\top},$
$\displaystyle\tau^{\beta_{i}/2}\,(\tau_{j}+s_{j})^{\beta_{j}/2}\colon$
$\displaystyle A_{1,i}\Sigma^{-1}A_{1,j}^{\top},$
$\displaystyle(\tau_{i}+t_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}\colon$
$\displaystyle A_{1,i}\Sigma^{-1}A_{1,j}^{\top},$
$\displaystyle\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}\colon$
$\displaystyle
B_{1,i}\Sigma^{-1}A_{1,j}+A_{1,i}^{\top}\Sigma^{-1}A_{1,j}^{\top}.$

Combining these, we find that the aggregate contribution of these terms to
$\mathcal{R}$ is

(94)
$\sum_{i,j\in\mathcal{F}}\Big{[}A_{6,i,j}+A_{1,i}\Sigma^{-1}A_{1,j}^{\top}\Big{]}\left((\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\right)\left((\tau_{j}+s_{j})^{\beta_{j}/2}-\tau_{j}^{\beta_{j}/2}\right).$

The significant $S$-terms appear only from (93). Their combined contribution
is

(95)
$\sum_{i=1}^{n}\Big{[}S_{\alpha_{i},A_{5,i}}(t_{i})+S_{\alpha_{i},A_{5,i}}(-s_{i})-S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})\Big{]}.$

The accumulated error is of order

$o\left(\sum_{i=1}^{n}\Big{[}t_{i}^{\beta_{i}}+s_{i}^{\beta_{i}}+\tau_{i}^{\beta_{i}}+|t_{i}|^{\alpha_{i}}+|s_{i}|^{\alpha_{i}}+|t_{i}-s_{i}|^{\alpha_{i}}\Big{]}\right).$

∎

####  8.3.6. Limiting process

######  Lemma 21.

Let $\Lambda>0$, $\bm{S}>\bm{0}$ and let

$Q_{u}=\left\\{\bm{\tau}\in\mathbb{R}^{n}_{+}\colon\tau_{i}=0\ \text{if}\
i\in\mathcal{I}^{c}\ \text{and}\ \tau_{i}\leq
u^{2/\alpha_{i}-2/\nu_{i}}\Lambda_{i}/S_{i}\ \text{if}\
i\in\mathcal{I}\right\\}.$

Then, the rescaled by $u^{-2/\bm{\nu}}$ contitional mean vector
$\bm{d}_{u,\bm{\tau}}$ converges uniformly

$\lim_{u\to\infty}\sup_{\bm{\tau}\in
Q_{u}}\sup_{\bm{t}\in[\bm{0},\bm{S}]}\left|u^{2}\,\bm{d}_{u,\bm{\tau}}\left(\bm{t}\right)-\bm{d}(\bm{t})\right|=0$

to the following limit

(96)
$\bm{d}(\bm{t})=\sum_{i\in\mathcal{I}\cup\mathcal{J}}S_{\alpha_{i},A_{5,i}}(t_{i})\,\bm{w}+\sum_{i\in\mathcal{J}\cup\mathcal{K}}A_{2,i}\,\bm{w}\,t_{i}^{\beta_{i}}.$

The rescaled conditional covariance matrix $\mathcal{R}_{u,\bm{\tau}}$
converges uniformly

$\lim_{u\to\infty}\sup_{\bm{\tau}\in
Q_{u}}\sup_{\bm{t},\,\bm{s}\in[\bm{0},\bm{S}]}\left\|u^{2}\,\mathcal{R}_{u,\bm{\tau}}\left(\bm{t},\bm{s}\right)-\mathcal{R}(\bm{t},\bm{s})\right\|=0$

to the following limit:

(97)
$\mathcal{R}(\bm{t},\bm{s})=\sum_{i\in\mathcal{I}\cup\mathcal{J}}R_{\alpha_{i},A_{5,i}}(t_{i},s_{i})+\sum_{i,j\in(\mathcal{J}\cup\mathcal{K})\cap\mathcal{F}}D_{i,j}\,t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2},$

where $R_{\alpha,V}$ is defined in (10) and $D_{i,j}$ in (81).

Assuming A2.6 and using Lemma 16 we find that the covariance (97) corresponds
to the following Gaussian random field:

$\bm{Y}(\bm{t})=\sum_{i\in\mathcal{I}\cup\mathcal{J}}\bm{Y}_{\alpha_{i},A_{5,i}}(t_{i})+\sum_{k\in\mathcal{F}}C_{k}(\bm{t})\,\bm{\mathcal{N}}_{k},\qquad
C_{k}(\bm{t})=\sum_{i\in\mathcal{F}}C_{i,k}\,t_{i}^{\beta_{i}/2},$

where

  1. (1)

$\bm{Y}_{\alpha_{i},A_{5,i}}$ are operator fractional Brownian motions
associated to $R_{\alpha_{i},A_{5,i}}$, independent of each other,

  2. (2)

$\bm{\mathcal{N}}_{k}$ are standard Gaussian vectors, independent of each
other and of $\bm{Y}_{\alpha_{i},A_{5,i}}$,

  3. (3)

$(C_{i,k})_{i,k\in\mathcal{F}}$, is a family of $d\times d$ matrices
satisfying (9), whose existence is guaranteed by A2.6 and Lemma 16.

###  8.4. Calculations from the log-layer lemma

###### Proof of (40).

By the definition of $\bm{\chi}_{u,\bm{k}}$ as the condional process

$\bm{\chi}_{u,\bm{k}}(\bm{t})=u\,\Big{(}\bm{X}_{u,\bm{k}}(\bm{t})-\bm{X}_{u,\bm{k}}(\bm{0})\
\Big{|}\ \bm{X}_{u,\bm{k}}(\bm{0})=u\bm{b}-\frac{\bm{x}}{u}\Big{)}$

we have

(98) $\displaystyle\mathbb{E}\left\\{\bm{\chi}_{u,\bm{k}}(\bm{t})\right\\}$
$\displaystyle=-u\Big{[}I-R_{u}(\bm{\Lambda}\bm{k}+\bm{t},\bm{\Lambda}\bm{k})\,\Sigma_{u}^{-1}(\bm{\Lambda}\bm{k})\Big{]}\left(u\,\bm{b}-\frac{\bm{x}}{u}\right)$
(99)
$\displaystyle=-u^{2}\,\bm{d}_{u,\bm{\Lambda}\bm{k}}(\bm{t})+\Big{[}I-R_{u}(\bm{\Lambda}\bm{k}+\bm{t},\bm{\Lambda}\bm{k})\,\Sigma_{u}^{-1}(\bm{\Lambda}\bm{k})\Big{]}\bm{x}.$

By 87, we have that for every $\varepsilon>0$ exists $\delta>0$ such that if
$|\bm{\tau}|,\,|\bm{t}|<\delta$, then

$\left\|I-R(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\right\|\leq\varepsilon$

Setting $\bm{\tau}\leadsto u^{-2/\bm{\nu}}\bm{\Lambda}\bm{k}$ and
$\bm{t}\leadsto u^{-2/\bm{\nu}}\bm{t}$, with $\bm{k}$ now belonging to $Q_{u}$
defined by (7.1), and noting that both belong to a shrinking, as $u\to\infty$,
vicinity of zero, we obtain the following result: for every $\varepsilon>0$
there exists $u_{0}$ such that for all $u\geq u_{0}$ holds

$\left|-u\Big{[}I-R(\bm{\tau}+\bm{t},\bm{\tau})\,\Sigma^{-1}(\bm{\tau})\Big{]}\,\frac{\bm{x}}{u}\right|\leq\varepsilon\,\sum_{j=1}^{d}|x_{j}|$

Next, we want to study the other term of (98). By (86),

$\bm{d}_{\,\bm{\tau}}(\bm{t})\sim\sum_{i=1}^{n}\Bigg{[}A_{2,i}\Big{[}(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\Big{]}+S_{\alpha_{i},A_{5,i}}(t_{i})\Bigg{]}\bm{w}\\\\[7.0pt]
+\sum_{i,j\in\mathcal{F}}\tau_{j}^{\beta_{j}/2}\,\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}\Big{[}A_{6,i,j}+A_{1,i}\Sigma^{-1}A_{1,j}^{\top}\Big{]}\bm{w}+\epsilon(\bm{\tau},\bm{t})$

with $\epsilon$ satisfying the following condition:

(100) $\begin{gathered}\text{for every}\ \varepsilon>0,\ \text{there exists}\
\delta>0\ \text{such that if}\ |\bm{\tau}|,\,|\bm{t}|<\delta,\ \text{then}\\\
\left|\epsilon(\bm{\tau},\bm{t})\right|\leq\varepsilon\sum_{i=1}^{n}\Big{[}\tau_{i}^{\beta_{i}}+t_{i}^{\beta_{i}}+|t_{i}|^{\alpha_{i}}\Big{]}.\end{gathered}$

Bounding the mixed terms sum by

$2\,\tau_{j}^{\beta_{j}/2}\,\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}\leq\varepsilon\,\tau_{j}^{\beta_{j}}+\varepsilon^{-1}\,\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}^{2}$

with the same $\varepsilon$ as above, we obtain the following inequality:

$\left|\bm{d}_{\,\bm{\tau}}(\bm{t})\right|\leq
c\sum_{i=1}^{n}\left[\Big{[}(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\Big{]}+\varepsilon^{-1}\,\Big{[}(\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}^{2}+|t_{i}|^{\alpha_{i}}\right]+\varepsilon\sum_{i=1}^{n}\Big{[}\tau_{i}^{\beta_{i}}+t_{i}^{\beta_{i}}\Big{]}.$

Set again $\bm{\tau}\leadsto u^{-2/\bm{\nu}}\,\bm{\Lambda}\bm{k}$ and
$\bm{t}\leadsto u^{-2/\bm{\nu}}\,\bm{t}$. New $\bm{k}$ and $\bm{t}$ belong to
$Q_{u}$ and $[\bm{0},\bm{\Lambda}]$ correspondingly.

Using the following inequality

(101) $(x+1)^{\beta}-x^{\beta}\leq c\,(x\vee 1)^{\beta-1}\quad\text{if}\quad
x\geq 0,$

which is valid with some constant $c>0$, we obtain that

$(\Lambda_{i}\,k_{i}+t_{i})^{\zeta}-(\Lambda_{i}\,k_{i})^{\zeta}\leq\Big{[}(\Lambda_{i}\,k_{i}+\Lambda_{i})^{\zeta}-(\Lambda_{i}\,k_{i})^{\zeta}\Big{]}\leq
c_{2}\,\Lambda_{i}^{\zeta}\,(k_{i}\vee 1)^{\zeta-1}$

for $\zeta=\beta_{i}$ or $\zeta=\beta_{i}/2$. Therefore, the first condition
of (39) holds with

$G\coloneqq
c_{3}\sum_{i=1}^{n}\left[u^{2-2\beta_{i}/\nu_{i}}\Lambda_{i}^{\beta_{i}}\Big{[}(k_{i}\vee
1)^{\beta_{i}-1}+\varepsilon^{-1}\,(k_{i}\vee
1)^{\beta_{i}-2}+\varepsilon\left(k_{i}^{\beta_{i}}+1\right)\Big{]}+u^{2-2\alpha_{i}/\nu_{i}}\Lambda_{i}^{\alpha_{i}}\right]$

where $\varepsilon$ can be made as small as required by taking $u$ large
enough.

Next, we seek for a bound of the variance of
$\bm{w}_{F}^{\top}\,\bm{\chi}_{u,\bm{k},F}(\bm{t})$. We have:

$\operatorname{Var}\left\\{\bm{w}_{F}^{\top}\,\bm{\chi}_{u,\bm{k},F}(\bm{t})\right\\}=\operatorname{Var}\left\\{\sum_{j\in
F}w_{j}\chi_{u,\bm{k},j}(\bm{t})\right\\}=\sum_{j\in
F}w_{j}^{2}\left[K_{u,\bm{k}}(\bm{t},\bm{t})\right]_{jj},$

where $K_{u,\bm{k}}$ is the covariance of $\bm{\chi}_{u,\bm{k}}(\bm{t})$:

$K_{u,\bm{k}}(\bm{t},\bm{s})=\mathbb{E}\left\\{\Big{[}\bm{\chi}_{u,\bm{k}}(\bm{t})-\bm{d}_{u,\bm{k}}(\bm{t})\Big{]}{\Big{[}\bm{\chi}_{u,\bm{k}}(\bm{s})-\bm{d}_{u,\bm{k}}(\bm{s})\Big{]}}^{\top}\right\\}=u^{2}\,\mathcal{R}_{u,\bm{\Lambda}\bm{k}}(\bm{t},\bm{s}).$

By Lemma 20, we have that

$\mathcal{R}_{\bm{\tau}}(\bm{t},\bm{t})=\sum_{i=1}^{n}\Big{[}S_{\alpha_{i},A_{5,i}}(t_{i})+S_{\alpha_{i},A_{5,i}}(-t_{i})\Big{]}\\\\[7.0pt]
+\sum_{i,j\in\mathcal{F}}\Big{[}A_{6,i,j}+A_{1,i}\Sigma^{-1}A_{1,j}^{\top}\Big{]}\left((\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\right)\left((\tau_{j}+t_{j})^{\beta_{j}/2}-\tau_{j}^{\beta_{j}/2}\right)+\epsilon(\bm{\tau},\bm{t})$

with $\epsilon$ satisfying (100). Setting $\bm{\tau}\leadsto
u^{-2/\bm{\nu}}\bm{\Lambda}\bm{k}$ and $\bm{t}\leadsto u^{-2/\bm{\nu}}\bm{t}$,
and using again (101), we find that the second condition of (39) holds with

$\sigma^{2}\coloneqq
c_{4}\sum_{i=1}^{n}\left[u^{2-2\beta_{i}/\nu_{i}}\,\Lambda_{i}^{\beta_{i}}\Big{[}(k_{i}\vee
1)^{\beta_{i}-2}+\varepsilon\left(k_{i}^{\beta_{i}}+1\right)\Big{]}+u^{2-2\alpha_{i}/\nu_{i}}\Lambda_{i}^{\alpha_{i}}\right],$

where $c_{4}>0$ and $\varepsilon$ can be made arbitrarily small by taking $u$
to be large enough. Adding the two bounds together yields

$G+\sigma^{2}=c_{5}\sum_{i=1}^{n}\Bigg{[}u^{2-2\beta_{i}/\nu_{i}}\Lambda_{i}^{\beta_{i}}\Big{[}(k_{i}\vee
1)^{\beta_{i}-1}+\varepsilon^{-1}\,(k_{i}\vee
1)^{\beta_{i}-2}+\varepsilon\left(k_{i}^{\beta_{i}}+1\right)\Big{]}+u^{2-2\alpha_{i}/\nu_{i}}\Lambda_{i}^{\alpha_{i}}\Bigg{]}.$

∎

###  8.5. Calculations from the double sum lemma

The covariance function of the field $(\bm{X}(\bm{t})+\bm{X}(\bm{s}))/2$ is
given by:

$\displaystyle R(\bm{t}_{1},\bm{s}_{1},\bm{t}_{2},\bm{s}_{2})$
$\displaystyle=\frac{1}{4}\,\mathbb{E}\left\\{\Big{[}\bm{X}(\bm{t}_{1})+\bm{X}(\bm{s}_{1})\Big{]}\Big{[}\bm{X}(\bm{t}_{2})+\bm{X}(\bm{s}_{2})\Big{]}^{\top}\right\\}$
(102)
$\displaystyle=\frac{1}{4}\Big{[}R(\bm{t}_{1},\bm{t}_{2})+R(\bm{t}_{1},\bm{s}_{2})+R(\bm{s}_{1},\bm{t}_{2})+R(\bm{s}_{1},\bm{s}_{2})\Big{]}.$

####  8.5.1. Inverse of Sigma

By (102), we have

$\Sigma(\bm{t},\bm{s})=\frac{1}{4}\Big{[}\Sigma(\bm{t})+\Sigma(\bm{s})+R(\bm{t},\bm{s})+R(\bm{s},\bm{t})\Big{]}.$

Using (90), we can rewrite it as follows:

$\Sigma(\bm{t},\bm{s})=\Sigma-\frac{1}{4}\Big{[}C(\bm{t},\bm{t})+C(\bm{s},\bm{s})+C(\bm{t},\bm{s})+C(\bm{s},\bm{t})\Big{]}\eqqcolon\Sigma-B(\bm{t},\bm{s}),$

where we have introduced one more shorthand $B(\bm{t},\bm{s})$. Similarly to
what we did in Lemma 17,

(103)
$\Sigma^{-1}(\bm{t},\bm{s})-\Sigma^{-1}=\Sigma^{-1}B(\bm{t},\bm{s})\,\Sigma^{-1}+\Sigma^{-1}B(\bm{t},\bm{s})\,\Sigma^{-1}B(\bm{t},\bm{s})\,\Sigma^{-1}+\epsilon,$

where, similarly to (91), $\epsilon$ can be shown to satisfy

$\epsilon=O\left(\sum_{i=1}^{n}\Big{[}t_{i}^{3\beta_{i}}+s_{i}^{3\beta_{i}}+|t_{i}-s_{i}|^{3\alpha_{i}}\Big{]}\right).$

Using (92), we can find an expression for $B(\bm{t},\bm{s})$:

(104) $\displaystyle B(\bm{t},\bm{s})$
$\displaystyle\sim\frac{1}{4}\sum_{i=1}^{n}\Big{[}2\,B_{1,i}\,t_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,t_{i}^{\beta_{i}}+2\,B_{1,i}\,s_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,s_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})+S_{\alpha_{i},A_{5,i}}(s_{i}-t_{i})\Big{]}$
$\displaystyle\hskip
10.0pt+\frac{1}{4}\,\sum_{i,j\in\mathcal{F}}A_{6,i,j}\Big{[}t_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}\Big{]}$

with $B_{k,i}=A_{k,i}+A_{k,i}^{\top}$, $k=1,2$ and within the same magnitude
of error.

The quadratic term of (103) reads:

$B(\bm{t},\bm{s})\,\Sigma^{-1}B(\bm{t},\bm{s})\sim\frac{1}{4}\,\sum_{i,j\in\mathcal{F}}B_{1,i}\,\Sigma^{-1}B_{1,j}\Big{[}t_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}}\,t_{j}^{\beta_{j}/2}\Big{]}.$

Finally, we arrive at

$\displaystyle\Sigma^{-1}(\bm{t},\bm{s})-\Sigma^{-1}$
$\displaystyle\sim\frac{1}{4}\,\Sigma^{-1}\,\sum_{i=1}^{n}\Big{[}2\,B_{1,i}\,t_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,t_{i}^{\beta_{i}}+2\,B_{1,i}\,s_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,s_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})+S_{\alpha_{i},A_{5,i}}(s_{i}-t_{i})\Big{]}\Sigma^{-1}$
$\displaystyle\hskip
10.0pt+\frac{1}{4}\,\sum_{i,j\in\mathcal{F}}\Sigma^{-1}\Big{[}A_{6,i,j}+B_{1,i}\,\Sigma^{-1}B_{1,j}\Big{]}\,\Sigma^{-1}\Big{[}t_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}\Big{]}.$

Rewriting it in terms of
$\widetilde{D}_{i,j}=A_{6,i,j}+B_{1,j}\,\Sigma^{-1}B_{1,j}$ at

(105) $\displaystyle\Sigma^{-1}(\bm{t},\bm{s})-\Sigma^{-1}$
$\displaystyle=\frac{1}{4}\,\Sigma^{-1}\,\sum_{i=1}^{n}\Big{[}2\,B_{1,i}\,t_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,t_{i}^{\beta_{i}}+2\,B_{1,i}\,s_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,s_{i}^{\beta_{i}}$
$\displaystyle+S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})+S_{\alpha_{i},A_{5,i}}(s_{i}-t_{i})\Big{]}\Sigma^{-1}$
$\displaystyle\hskip
10.0pt+\frac{1}{4}\,\Sigma^{-1}\,\sum_{i,j\in\mathcal{F}}\widetilde{D}_{i,j}\,\Big{[}t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}+t_{i}^{\beta_{i}/2}\,t_{j}^{\beta_{j}/2}+s_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2}\Big{]}\Sigma^{-1}+\epsilon.$

The error term $\epsilon$ satisfies

(106)
$\epsilon=o\left(\sum_{i=1}^{n}\Big{[}t_{i}^{\beta_{i}}+s_{i}^{\beta_{i}}+|t_{i}-s_{i}|^{\alpha_{i}}\Big{]}\right).$

####  8.5.2. Exponential prefactor

Multiplying (105) by $\bm{b}$ on both sides, and using the fact that by A2.3

$\bm{w}^{\top}B_{1,i}\,\bm{w}=0,\quad\bm{w}^{\top}B_{2,i}\,\bm{w}=2\,\bm{w}^{\top}A_{2,i}\,\bm{w},\quad\bm{w}^{\top}\widetilde{D}_{i,j}\,\bm{w}=\Xi_{i,j},$

where $\Xi_{i,j}$ is defined in (81), and also

$\bm{w}^{\top}S_{\alpha,A}(t)\,\bm{w}=\bm{w}^{\top}A\,\bm{w}\,\mathbb{1}_{t\geq
0}\,|t|^{\alpha}+\underbracket{\bm{w}^{\top}A^{\top}\,\bm{w}}_{=\,\bm{w}^{\top}A\,\bm{w}}\,\mathbb{1}_{t<0}\,|t|^{\alpha}=\bm{w}^{\top}A\,\bm{w}\,|t|^{\alpha},$

we obtain

(107)
$\bm{b}^{\top}\Big{[}\Sigma^{-1}(\bm{\tau},\bm{\lambda})-\Sigma^{-1}\Big{]}\bm{b}\sim\sum_{i=1}^{n}\left[\bm{w}^{\top}A_{2,i}\,\bm{w}\,\Big{[}\tau_{i}^{\beta_{i}}+\lambda_{i}^{\beta_{i}}\Big{]}+\frac{\bm{w}^{\top}A_{5,i}\,\bm{w}}{2}\,\left|\lambda_{i}-\tau_{i}\right|^{\alpha_{i}}\right]\\\\[7.0pt]
+\frac{1}{4}\,\sum_{i,j\in\mathcal{F}}\Xi_{i,j}\Big{[}\tau_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}+\lambda_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}+\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}+\lambda_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}\Big{]}$

with the same error as in (106).

####  8.5.3. Conditional mean: formula

The conditional mean vector $\bm{d}_{\bm{\tau},\bm{\lambda}}(\bm{t},\bm{s})$
of the field $(\bm{X}(\bm{t})+\bm{X}(\bm{s}))/2$ is given by

$\bm{d}_{\bm{\tau},\bm{\lambda}}(\bm{t},\bm{s})=\Big{[}I-R(\bm{\tau}+\bm{t},\bm{\lambda}+\bm{s},\bm{\tau},\bm{\lambda})\,\Sigma^{-1}(\bm{\tau},\bm{\lambda})\Big{]}\bm{b},$

where $R(\bm{t}_{1},\bm{s}_{1},\bm{t}_{2},\bm{s}_{2})$ is defined by (102).

As in the proof of 19, to simplify the computations define $C$ and $B$ by

$R(\bm{\tau}+\bm{t},\bm{\lambda}+\bm{s},\bm{\tau},\bm{\lambda})\eqqcolon\Sigma-C,\qquad\Sigma^{-1}(\bm{\tau},\bm{\lambda})\eqqcolon\Sigma^{-1}+\Sigma^{-1}\,B\,\Sigma^{-1}.$

With these shorthands, we have

(108)
$I-R(\bm{\tau}+\bm{t},\bm{\lambda}+\bm{s},\bm{\tau},\bm{\lambda})\,\Sigma^{-1}(\bm{\tau},\bm{\lambda})=I-\Big{[}\Sigma-C\Big{]}\Big{[}\Sigma^{-1}+\Sigma^{-1}B\,\Sigma^{-1}\Big{]}=(C-B)\,\Sigma^{-1}+C\,\Sigma^{-1}B\,\Sigma^{-1}.$

By (103),

$C=\frac{1}{4}\Big{[}C(\bm{\tau}+\bm{t},\bm{\tau})+C(\bm{\tau}+\bm{t},\bm{\lambda})+C(\bm{\lambda}+\bm{s},\bm{\tau})+C(\bm{\lambda}+\bm{s},\bm{\lambda})\Big{]}$

with

$C(\bm{t},\bm{s})\sim\sum_{i=1}^{n}\Big{[}A_{1,i}\,t^{\beta_{i}^{\prime}}+A_{2,i}\,t^{\beta_{i}}+A_{1,i}^{\top}\,s_{i}^{\beta_{i}^{\prime}}+A_{2,i}^{\top}\,s_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})\Big{]}+\sum_{i,\,j\in\mathcal{F}}A_{6,i,j}\,t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2},$

and

$B\sim\frac{1}{4}\,\sum_{i=1}^{n}\Big{[}2\,B_{1,i}\,\tau_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,\tau_{i}^{\beta_{i}}+2\,B_{1,i}\,\lambda_{i}^{\beta_{i}^{\prime}}+2\,B_{2,i}\,\lambda_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(\tau_{i}-\lambda_{i})+S_{\alpha_{i},A_{5,i}}(\tau_{i}-\lambda_{i})\Big{]}\\\\[7.0pt]
+\frac{1}{4}\,\sum_{i,j\in\mathcal{F}}\widetilde{D}_{i,j}\,\Big{[}\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}+\tau_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}+\lambda_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}+\lambda_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}\Big{]}$

Let us calculate the leading order coefficients in (108). First, the $S$-type
contributions are

(109)
$F_{1}\coloneqq\frac{1}{4}\sum_{i=1}^{n}\Big{[}2\,S_{\alpha_{i},A_{5,i}}(t_{i})+2\,S_{\alpha_{i},A_{5,i}}(s_{i})+S_{\alpha_{i},A_{5,i}}(\tau_{i}+t_{i}-\lambda_{i})\\\\[7.0pt]
+S_{\alpha_{i},A_{5,i}}(\lambda_{i}+s_{i}-\tau_{i})-S_{\alpha_{i},A_{5,i}}(\tau_{i}-\lambda_{i})-S_{\alpha_{i},A_{5,i}}(\lambda_{i}-\tau_{i})\Big{]}\Sigma^{-1}.$

Next, the leading power-type orders give

$\displaystyle(\tau_{i}+t_{i})^{\beta_{i}^{\prime}}-\tau_{i}^{\beta_{i}^{\prime}},\
(\lambda_{i}+s_{i})^{\beta_{i}^{\prime}}-\lambda_{i}^{\beta_{i}}\colon\quad\frac{1}{2}\,A_{1,i},$
$\displaystyle(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}},\
(\lambda_{i}+s_{i})^{\beta_{i}}-\lambda_{i}^{\beta_{i}}\colon\quad\frac{1}{2}\,A_{2,i}.$

It remains to compute the mixed terms: from $C-D$:

(110)
$\begin{gathered}(\tau_{i}+t_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad(\lambda_{i}+s_{i})^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\\\
(\tau_{i}+t_{i})^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\quad(\lambda_{i}+s_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2}\end{gathered}\colon\quad\frac{1}{4}\,A_{6,i,j}$

and

(111)
$\begin{gathered}\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad\tau_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\\\
\lambda_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad\lambda_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}\end{gathered}\colon\quad-\frac{1}{4}\,\widetilde{D}_{i,j}.$

From $C\,\Sigma^{-1}B$:

(112)
$\begin{gathered}(\tau_{i}+t_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad(\tau_{i}+t_{i})^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\\\
(\lambda_{i}+s_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad(\lambda_{i}+s_{i})^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}\end{gathered}\colon\quad\frac{1}{4}\,A_{1,i}\,\Sigma^{-1}B_{1,j}$

and

(113)
$\begin{gathered}\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad\tau_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\\\
\lambda_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad\lambda_{i}^{\beta_{i}/2},\,\lambda_{j}^{\beta_{j}/2}\end{gathered}\colon\quad\frac{1}{4}\,A_{1,i}^{\top}\,\Sigma^{-1}B_{1,j}.$

Combining (110) and (112) gives

$\begin{gathered}(\tau_{i}+t_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad(\tau_{i}+t_{i})^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\\\
(\lambda_{i}+s_{i})^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad(\lambda_{i}+s_{i})^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2}\end{gathered}\colon\quad\frac{1}{4}\,A_{6,i,j}+\frac{1}{4}\,A_{1,i}\,\Sigma^{-1}B_{1,j}.$

Similarly for (111) and (113):

$\begin{gathered}\tau_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad\tau_{i}^{\beta_{i}/2}\,\lambda_{j}^{\beta_{j}/2},\\\
\lambda_{i}^{\beta_{i}/2}\,\tau_{j}^{\beta_{j}/2},\quad\lambda_{i}^{\beta_{i}/2},\,\lambda_{j}^{\beta_{j}/2}\end{gathered}\colon\quad-\frac{1}{4}\,A_{6,i,j}-\frac{1}{4}\,A_{1,i}\,\Sigma^{-1}B_{1,j},$

where we have used

$-\widetilde{D}_{i,j}+A_{1,i}^{\top}\,\Sigma^{-1}B_{1,j}=-A_{6,i,j}-B_{1,i}\,\Sigma^{-1}B_{1,j}+A_{1,i}^{\top}\,\Sigma^{-1}B_{1,j}=-A_{6,i,j}-A_{1,i}\,\Sigma^{-1}B_{1,j}$

The aggregate power-type contribution is

(114) $\displaystyle F_{2}$
$\displaystyle\coloneqq\frac{1}{2}\,\sum_{i=1}^{n}\Bigg{[}A_{1,i}\left[\Big{(}(\tau_{i}+t_{i})^{\beta_{i}^{\prime}}-\tau_{i}^{\beta_{i}^{\prime}}\Big{)}+\Big{(}(\lambda_{i}+s_{i})^{\beta_{i}^{\prime}}-\lambda_{i}^{\beta_{i}^{\prime}}\Big{)}\right]$
$\displaystyle+A_{2,i}\left[\Big{(}(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\Big{)}+\Big{(}(\lambda_{i}+s_{i})^{\beta_{i}}-\lambda_{i}^{\beta_{i}}\Big{)}\right]\Bigg{]}\Sigma^{-1}$
$\displaystyle\hskip
10.0pt+\frac{1}{4}\sum_{i,j\in\mathcal{F}}\Big{[}A_{6,i,j}+A_{1,i}\,\Sigma^{-1}B_{1,j}\Big{]}\Sigma^{-1}\left(\tau_{j}^{\beta_{j}/2}+\lambda_{j}^{\beta_{j}/2}\right)$
$\displaystyle\times\Bigg{[}\left((\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\right)\,+\left((\lambda_{i}+s_{i})^{\beta_{i}/2}-\lambda_{i}^{\beta_{i}/2}\right)\,\Bigg{]}$

We have thus shown that

(115)
$I-R(\bm{\tau}+\bm{t},\bm{\lambda}+\bm{s},\bm{\tau},\bm{\lambda})\,\Sigma^{-1}(\bm{\tau},\bm{\lambda})=F_{1}+F_{2}+\epsilon$

with $F_{2}$ defined in (114), $F_{1}$ defined in (109) and the error
$\epsilon$ of order

(116)
$o\Bigg{(}\sum_{i=1}^{n}\Big{[}\tau_{i}^{\beta_{i}}+\lambda_{i}^{\beta_{i}}+t_{i}^{\beta_{i}}+s_{i}^{\beta_{i}}+|t_{i}|^{\alpha_{i}}+|s_{i}|^{\alpha_{i}}+|t_{i}-s_{i}|^{\alpha_{i}}+|\tau_{i}+t_{i}-\lambda_{i}|^{\alpha_{i}}+|\lambda_{i}+s_{i}-\tau_{i}|^{\alpha_{i}}\Big{]}\Bigg{)}.$

####  8.5.4. Conditional mean: upper bound

Recall that we are interested in the upper bound for the rescaled conditional
mean vector $\bm{d}_{u,\bm{\tau},\bm{\lambda}}$ uniform in
$\bm{t},\,\bm{s}\in[\bm{0},\bm{S}]$ and $\bm{\tau}$ and $\bm{\lambda}$ such
that with some $\mathcal{I}^{\prime}\subset\mathcal{I}$

(117) $\bm{0}\leq\bm{\tau}\leq\bm{\lambda}\leq
u^{2/\bm{\nu}-2/\bm{\beta}}\bm{\Lambda}/\bm{S},\quad\bm{\tau}_{\mathcal{J}\cup\mathcal{K}}=\bm{\lambda}_{\mathcal{J}\cup\mathcal{K}}=\bm{0}_{\mathcal{J}\cup\mathcal{K}},\quad\bm{\tau}_{\mathcal{I}^{\prime}}=\bm{\lambda}_{\mathcal{I}^{\prime}},\quad\bm{\tau}_{\mathcal{I}\setminus\mathcal{I}^{\prime}}+\mathcal{S}_{\mathcal{I}\setminus\mathcal{I}^{\prime}}<\bm{\lambda}_{\mathcal{I}^{\prime}\setminus\mathcal{I}}.$

Let us bound $F_{1}\,\bm{w}$ and $F_{2}\,\bm{w}$ separately.

Bound for $F_{1}\,\bm{w}$. Multiplying $F_{1}$ by $\bm{w}$ on the right and
rescaling all time parameters by $u^{-2/\bm{\nu}}$, we find that

$u^{2}\,F_{1,u}\,\bm{w}\coloneqq\frac{1}{4}\sum_{i=1}^{n}u^{2-2\alpha_{i}/\nu_{i}}\Big{[}2\,S_{\alpha_{i},A_{5,i}}(t_{i})+2\,S_{\alpha_{i},A_{5,i}}(s_{i})+S_{\alpha_{i},A_{5,i}}(\tau_{i}+t_{i}-\lambda_{i})\\\\[7.0pt]
+S_{\alpha_{i},A_{5,i}}(\lambda_{i}+s_{i}-\tau_{i})-S_{\alpha_{i},A_{5,i}}(\tau_{i}-\lambda_{i})-S_{\alpha_{i},A_{5,i}}(\lambda_{i}-\tau_{i})\Big{]}\,\bm{w}.$

Then, with some $c_{1}>0$ holds

$\left|S_{\alpha_{i},A_{5,i}}(t_{i})\,\bm{w}\right|,\
\left|S_{\alpha_{i},A_{5,i}}(s_{i})\,\bm{w}\right|\leq
c_{1}\,S_{i}^{\alpha_{i}}$

and, using

(118) $\left|\left|x\pm 1\right|^{\zeta}-x^{\zeta}\right|\leq c_{2}\,(x\vee
1)^{\zeta-1}\quad\text{for}\quad x\geq 0,$

we find that

$\displaystyle\left|\Big{[}S_{\alpha_{i},A_{5,i}}(\tau_{i}+t_{i}-\lambda_{i})-S_{\alpha_{i},A_{5,i}}(\tau_{i}-\lambda_{i})\Big{]}\,\bm{w}\right|$
$\displaystyle\leq
c_{3}\Big{[}(\lambda_{i}-\tau_{i}-t_{i})^{\alpha_{i}}-(\lambda_{i}-\tau_{i})^{\alpha_{i}}\Big{]}$
$\displaystyle\leq c_{4}\,S_{i}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}-1},$
$\displaystyle\left|\Big{[}S_{\alpha_{i},A_{5,i}}(\lambda_{i}+s_{i}-\tau_{i})-S_{\alpha_{i},A_{5,i}}(\lambda_{i}-\tau_{i})\Big{]}\,\bm{w}\right|$
$\displaystyle\leq
c_{5}\Big{[}(\lambda_{i}-\tau_{i}+s_{i})^{\alpha_{i}}-(\lambda_{i}-\tau_{i})^{\alpha_{i}}\Big{]}$
$\displaystyle\leq c_{6}\,S_{i}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}-1}.$

Bound for $F_{2}\,\bm{w}$. Multiplying $F_{2}$ by $\bm{w}$ on the right,
rescaling all time parameters by $u^{-2/\bm{\nu}}$ and using
$A_{1,i}\,\bm{w}=\bm{0}$ yields

(119)
$u^{2}\,F_{2,u}\,\bm{w}=\sum_{i=1}^{n}u^{2-2\beta_{i}/\nu_{i}}\,B_{i}+\sum_{i,j\in\mathcal{F}}u^{2-\beta_{i}/\nu_{i}-\beta_{j}/\nu_{j}}\,B_{i,j},$

where we have introduced two shorthands

$\displaystyle
B_{i}\coloneqq\frac{1}{2}\,A_{2,i}\,\bm{w}\left[\Big{(}(\tau_{i}+t_{i})^{\beta_{i}}-\tau_{i}^{\beta_{i}}\Big{)}+\Big{(}(\lambda_{i}+s_{i})^{\beta_{i}}-\lambda_{i}^{\beta_{i}}\Big{)}\right]$
$\displaystyle
B_{i,j}\coloneqq\frac{1}{4}\,D_{i,j}\,\bm{w}\left[\left((\tau_{i}+t_{i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\right)\,+\left((\lambda_{i}+s_{i})^{\beta_{i}/2}-\lambda_{i}^{\beta_{i}/2}\right)\,\right]\left(\tau_{i}^{\beta_{j}/2}+\lambda_{j}^{\beta_{j}/2}\right).$

They can all be bounded using (101) as follows:

$\left|B_{i}\right|\leq c_{1}S_{i}\,(\lambda_{i}\vee
S_{i})^{\beta_{i}-1},\quad\left|B_{i,j}\right|\leq
c_{1}\Big{[}S_{i}\,(\tau_{i}\vee S_{i})^{\beta_{i}-1}+S_{j}\,(\lambda_{j}\vee
S_{j})^{\beta_{j}-1}\Big{]}\,\left(\tau_{j}^{\beta_{j}/2}+\lambda_{j}^{\beta_{j}/2}\right).$

To simplify the bound further, let us get rid of the mixing, applying

(120) $2\,x\,y\leq\varepsilon x+\varepsilon^{-1}y$

to each of the terms. Hence, the right-hand side of (119) is at most

$c_{1}\sum_{i=1}^{n}u^{2-2\beta_{i}/\nu_{i}}\Big{[}S_{i}\,(\lambda_{i}\vee
S_{i})^{\beta_{i}-1}+\varepsilon\,\lambda_{j}^{\beta_{i}}+\varepsilon^{-1}\,S_{j}^{2}\,(\lambda_{j}\vee
S_{j})^{\beta_{j}-2}\Big{]}.$

Error. The error (116) is no larger than

$\varepsilon\sum_{i=1}^{n}\left[u^{2-2\beta_{i}/\nu_{i}}\,(\lambda_{i}\vee
S_{i})^{\beta_{i}}+u^{2-2\alpha_{i}/\nu_{i}}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}}\right],$

where $\varepsilon$ can be made arbitrarily small by choosing $u$ large
enough.

Combined bound.

$\left|u^{2}\,\bm{d}_{u,\bm{\tau},\bm{\lambda}}(\bm{t},\bm{s})\right|\leq
c_{1}\sum_{i=1}^{n}\Bigg{[}u^{2-2\alpha_{i}/\nu_{i}}\,\Big{[}S_{i}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}-1}+\varepsilon\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}}\Big{]}\\\\[7.0pt]
+u^{2-2\beta_{i}/\nu_{i}}\,\Big{[}S_{i}\,(\lambda_{i}\vee
S_{i})^{\beta_{i}-1}+\varepsilon\,(\lambda_{i}\vee
S_{i})^{\beta_{i}}+\varepsilon^{-1}\,S_{j}^{2}\,(\lambda_{j}\vee
S_{j})^{\beta_{j}-2}\Big{]}\Bigg{]}$

The right-hand side of this bound can be taken as $G$ for (52).

####  8.5.5. Conditional covariance

Finally, we need a bound for

$\mathcal{R}_{\bm{\tau},\bm{\lambda}}(\bm{t}_{1},\bm{s}_{1},\bm{t}_{2},\bm{s}_{2})=R(\bm{\tau}+\bm{t}_{1},\bm{\lambda}+\bm{s}_{1},\bm{\tau}+\bm{t}_{2},\bm{\lambda}+\bm{s}_{2})\\\
+R(\bm{\tau}+\bm{t}_{1},\bm{\lambda}+\bm{s}_{1},\bm{\tau},\bm{\lambda})\,\Sigma^{-1}(\bm{\tau},\bm{\lambda})\,R(\bm{\tau},\bm{\lambda},\bm{\tau}+\bm{t}_{2},\bm{\lambda}+\bm{s}_{2}).$

Introduce the following shorthands:

$\displaystyle
R(\bm{\tau}+\bm{t}_{1},\bm{\lambda}+\bm{s}_{1},\bm{\tau}+\bm{t}_{2},\bm{\lambda}+\bm{s}_{2})\eqqcolon\Sigma-
C_{1},$ $\displaystyle
R(\bm{\tau}+\bm{t}_{1},\bm{\lambda}+\bm{s}_{1},\bm{\tau},\bm{\lambda})\eqqcolon\Sigma-
C_{2},$ $\displaystyle
R(\bm{\tau},\bm{\lambda},\bm{\tau}+\bm{t}_{2},\bm{\lambda}+\bm{s}_{2})\eqqcolon\,\Sigma-
C_{3}$
$\displaystyle\Sigma^{-1}(\bm{\tau},\bm{\lambda})\eqqcolon\,\Sigma^{-1}+\Sigma^{-1}B\,\Sigma^{-1}\,+\Sigma^{-1}B\,\Sigma^{-1}B\,\Sigma^{-1},$

Using these shorthands, we find that $\mathcal{R}_{\bm{\tau},\bm{\lambda}}$
satisfies

$\mathcal{R}_{\bm{\tau},\bm{\lambda}}(\bm{t}_{1},\bm{s}_{1},\bm{t}_{2},\bm{s}_{2})\sim
G_{1}+G_{2},\quad\text{where}\quad\begin{cases}G_{1}\coloneqq-
C_{1}+C_{2}-B+C_{3},\\\\[3.0pt] \begin{multlined}G_{2}\coloneqq
B\,\Sigma^{-1}C_{3}+C_{2}\,\Sigma^{-1}B\\\\[-10.0pt]
-C_{2}\,\Sigma^{-1}C_{3}-B\,\Sigma^{-1}B.\end{multlined}G_{2}\coloneqq
B\,\Sigma^{-1}C_{3}+C_{2}\,\Sigma^{-1}B\\\\[-10.0pt]
-C_{2}\,\Sigma^{-1}C_{3}-B\,\Sigma^{-1}B.\end{cases}$

We shall also need the following formula:

$C(\bm{t},\bm{s})\sim\sum_{i=1}^{n}\Big{[}A_{1,i}\,t^{\beta_{i}^{\prime}}+A_{2,i}\,t^{\beta_{i}}+A_{1,i}^{\top}\,s_{i}^{\beta_{i}^{\prime}}+A_{2,i}^{\top}\,s_{i}^{\beta_{i}}+S_{\alpha_{i},A_{5,i}}(t_{i}-s_{i})\Big{]}+\sum_{i,\,j\in\mathcal{F}}A_{6,i,j}\,t_{i}^{\beta_{i}/2}\,s_{j}^{\beta_{j}/2},$

Terms with $(\bm{t}_{1},\bm{t}_{2})$. In $G_{1}$, these terms come from

$G_{1,1}=\frac{1}{4}\Big{[}-C(\bm{\tau}+\bm{t}_{1},\bm{\tau}+\bm{t}_{2})+C(\bm{\tau}+\bm{t}_{1},\bm{\tau})-C(\bm{\tau},\bm{\tau})+C(\bm{\tau},\bm{\tau}+\bm{t}_{2})\Big{]}.$

The following coefficients are zero:

$(\tau_{i}+t_{1,i})^{\beta_{i}^{\prime}},\ (\tau_{i}+t_{1,i})^{\beta_{i}},\
(\tau_{i}+t_{2,i})^{\beta_{i}^{\prime}},\ (\tau_{i}+t_{2,i})^{\beta_{i}},\
\tau_{i}^{\beta_{i}^{\prime}},\ \tau_{i}^{\beta_{i}}.$

Mixed terms from $G_{1,1}$ and $G_{2}$:

$\frac{1}{4}\sum_{i,j\in\mathcal{F}}D_{i,j}\Big{[}(\tau_{i}+t_{1,i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}\Big{[}(\tau_{j}+t_{1,j})^{\beta_{j}/2}-\tau_{j}^{\beta_{j}/2}\Big{]}.$

$S$-terms:

$\frac{1}{4}\sum_{i=1}^{n}\Big{[}-S_{\alpha_{i},A_{5,i}}(t_{1,i}-t_{2,i})+S_{\alpha_{i},A_{5,i}}(t_{1,i})+S_{\alpha_{i},A_{5,i}}(-t_{2,i})\Big{]}.$

Rescaling everything by $u^{-2/\bm{\nu}}$, we obtain

$u^{2}\,\left\|\begin{subarray}{c}\text{terms with}\\\ \bm{t}_{1}\ \text{and}\
\bm{t}_{2}\end{subarray}\right\|\leq
c_{1}\sum_{i=1}^{n}u^{2-2\alpha_{i}/\nu_{i}}\,S_{i}^{\alpha_{i}}+c_{1}\sum_{i,j\in\mathcal{F}}u^{2-\beta_{i}/\nu_{i}-\beta_{j}/\nu_{j}}\,S_{i}\,(\tau_{i}\vee
S_{i})^{\beta_{i}/2-1}\,S_{j}\,(\tau_{j}\vee S_{j})^{\beta_{j}/2-1}.$

We can also simplify the bound by getting rid of the mixing:

$u^{2}\,\left\|\begin{subarray}{c}\text{terms with}\\\ \bm{t}_{1}\ \text{and}\
\bm{t}_{2}\end{subarray}\right\|\leq
c_{2}\sum_{i=1}^{n}\left[u^{2-2\alpha_{i}/\nu_{i}}\,S_{i}^{\alpha_{i}}+u^{2-2\beta_{i}/\nu_{i}}S_{i}^{2}\,(\tau_{i}\vee
S_{i})^{\beta_{i}-2}\right].$

Terms with $(\bm{t}_{1},\bm{s}_{2})$. In $G_{1}$, these terms come from

$G_{1,2}=\frac{1}{4}\Big{[}-C(\bm{\tau}+\bm{t}_{1},\bm{\lambda}+\bm{s}_{2})+C(\bm{\tau}+\bm{t}_{1},\bm{\lambda})-C(\bm{\lambda},\bm{\lambda})+C(\bm{\tau},\bm{\lambda}+\bm{s}_{2})\Big{]}.$

The following coefficients are zero:

$(\tau_{i}+t_{1,i})^{\beta_{i}^{\prime}},\ (\tau_{i}+t_{1,i})^{\beta_{i}},\
(\lambda_{i}+s_{2,i})^{\beta_{i}^{\prime}},\
(\lambda_{i}+s_{2,i})^{\beta_{i}},\ \lambda_{i}^{\beta_{i}^{\prime}},\
\lambda_{i}^{\beta_{i}}.$

Mixed terms from $G_{1,1}$ and $G_{2}$:

$\frac{1}{4}\sum_{i,j\in\mathcal{F}}D_{i,j}\Big{[}(\tau_{i}+t_{1,i})^{\beta_{i}/2}-\tau_{i}^{\beta_{i}/2}\Big{]}\Big{[}(\lambda_{j}+s_{1,j})^{\beta_{j}/2}-\lambda_{j}^{\beta_{j}/2}\Big{]}.$

$S$-terms:

$\frac{1}{4}\sum_{i=1}^{n}\Big{[}-S_{\alpha_{i},A_{5,i}}(\tau_{i}+t_{1,i}-\lambda_{i}-s_{2,i})+S_{\alpha_{i},A_{5,i}}(\tau_{i}+t_{1,i})-S_{\alpha_{i},A_{5,i}}(\tau_{i}-\lambda_{i})+S_{\alpha_{i},A_{5,i}}(-\lambda_{i}-s_{2,i})\Big{]}.$

Rescaling everything by $u^{-2/\bm{\nu}}$, and getting rid of the mixed terms
as above, we obtain

$u^{2}\,\left\|\begin{subarray}{c}\text{terms with}\\\ \bm{t}_{1}\ \text{and}\
\bm{t}_{2}\end{subarray}\right\|\leq
c_{1}\sum_{i=1}^{n}\left[u^{2-2\alpha_{i}/\nu_{i}}\,S_{i}\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}-1}+u^{2-2\beta_{i}/\nu_{i}}\,S_{i}^{2}\,(\lambda_{i}\vee
S_{i})^{\beta_{i}-2}.\right]$

The remaining terms (with $(\bm{s}_{1},\bm{t}_{2})$ and
$(\bm{s}_{1},\bm{s}_{2})$) may be estimated similarly.

Error. The accumulated error is at most

$\varepsilon\sum_{i=1}^{n}\Bigg{[}u^{2-2\beta_{i}/\nu_{i}}(\lambda_{i}\vee
S_{i})^{\beta_{i}}+u^{2-2\alpha_{i}/\nu_{i}}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}}\Bigg{]}$

with $\varepsilon$, which can be made arbitrarily small by taking $u$ large
enough.

Combined bound. Combining the bounds together, we find that

$u^{2}\,\left\|\mathcal{R}_{u,\bm{\tau},\bm{\lambda}}(\bm{t}_{1},\bm{s}_{1},\bm{t}_{1},\bm{s}_{1})\right\|\leq
c_{1}\sum_{i=1}^{n}\Bigg{[}u^{2-2\alpha_{i}/\nu_{i}}\Big{[}S_{i}^{\alpha_{i}}+S_{i}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}-1}+\varepsilon\,(\lambda_{i}-\tau_{i})^{\alpha_{i}}\Big{]}\\\\[7.0pt]
+u^{2-2\beta_{i}/\nu_{i}}\,\Big{[}S_{i}(\lambda_{i}\vee
S_{i})^{\beta_{i}-1}+\varepsilon(\lambda_{i}\vee
S_{i})^{\beta_{i}}\Big{]}\Bigg{]}$

We have therefore obtained that (52) holds with

$G+\sigma^{2}=c_{1}\sum_{i=1}^{n}\Bigg{[}u^{2-2\alpha_{i}/\nu_{i}}\,\Big{[}S_{i}^{\alpha_{i}}+S_{i}\,\left((\lambda_{i}-\tau_{i})\vee
S_{i}\right)^{\alpha_{i}-1}+\varepsilon(\lambda_{i}-\tau_{i})^{\alpha_{i}}\Big{]}\\\\[7.0pt]
+u^{2-2\beta_{i}/\nu_{i}}\,\Big{[}S_{i}\,(\lambda_{i}\vee
S_{i})^{\beta_{i}-1}+\varepsilon\,(\lambda_{i}\vee
S_{i})^{\beta_{i}}+\varepsilon^{-1}\,S_{j}^{2}\,(\lambda_{j}\vee
S_{j})^{\beta_{j}-2}\Big{]}\Bigg{]}.$

###  8.6. Integral estimate

###### Proof of Lemma 10.

Define a collection of sets
$\Omega_{F}=\left\\{\bm{x}\in\mathbb{R}^{d}\colon\bm{x}_{F}>\bm{0},\
\bm{x}_{F^{c}}<\bm{0}\right\\}$ indexed by $F\subset\\{1,\ldots,d\\}$ and
split the integral:

$\int_{\mathbb{R}^{d}}e^{\bm{w}^{\top}\bm{x}}\,\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{\chi}_{\bm{x}}(\bm{t})>\bm{x}\right\\}\mathop{d\bm{x}}=\sum_{F\in
2^{d}}\int_{\Omega_{F}}e^{\bm{w}^{\top}\bm{x}}\,\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{\chi}_{\bm{x}}(\bm{t})>\bm{x}\right\\}\mathop{d\bm{x}}.$

For $\bm{x}\in\Omega_{F}$ the probability under the integral may be bounded as
follows:

$\displaystyle\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{\chi}_{\bm{x}}(\bm{t})>\bm{x}\right\\}$
$\displaystyle\hskip
30.0pt\leq\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{w}_{F}^{\top}\big{(}\bm{\chi}_{\bm{x},F}(\bm{t})-\mathbb{E}\left\\{\bm{\chi}_{\bm{x},F}(\bm{t})\right\\}\big{)}>\bm{w}_{F}^{\top}\bm{x}_{F}-\bm{w}_{F}^{\top}\mathbb{E}\left\\{\bm{\chi}_{\bm{x},F}(\bm{t})\right\\}\right\\}$
$\displaystyle\hskip
30.0pt\leq\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{w}_{F}^{\top}\big{(}\bm{\chi}_{\bm{x}}-\mathbb{E}\left\\{\bm{\chi}_{\bm{x}}(\bm{t})\right\\}\big{)}>\bm{w}_{F}^{\top}\bm{x}_{F}-G-\varepsilon\sum_{j=1}^{d}|x_{j}|\right\\}$
$\displaystyle\hskip
30.0pt=\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\eta_{\bm{x},F}(\bm{t})>r_{F,\varepsilon}(\bm{x})-G\right\\},$

where

$r_{F,\varepsilon}(\bm{x})=\bm{w}_{F}^{\top}\,\bm{x}_{F}-\varepsilon\sum_{j=1}^{d}|x_{j}|\qquad\text{and}\qquad\eta_{\bm{x},F}(\bm{t})=\bm{w}_{F}^{\top}\big{(}\bm{\chi}_{\bm{x},F}(\bm{t})-\mathbb{E}\left\\{\bm{x}_{\bm{x},F}(\bm{t})\right\\}\big{)}.$

Let us split the domain $\Omega_{F}$ into two parts

$\Omega_{F,+}=\left\\{\bm{x}\in\Omega_{F}\colon
r_{F,\varepsilon}(\bm{x})>G\right\\}\qquad\text{and}\qquad\Omega_{F,-}=\Omega_{F}\setminus\Omega_{F,+}.$

Let us first deal with the integral over $\Omega_{F,-}$. It follows from
$\bm{w}_{F}^{\top}\,\bm{x}_{F}-\varepsilon\sum_{j=1}^{d}|x_{j}|<G$ that

$\sum_{j\in F}(w_{i}-\varepsilon)|x_{j}|-\varepsilon\sum_{j\in
F^{c}}|x_{j}|<G$

or, with $w_{*}=\min_{j\in F}w_{j}>0$ and $\varepsilon<w_{*}$,

$\varepsilon\sum_{j\in F}|x_{j}|\leq\frac{\varepsilon
G}{w_{*}-\varepsilon}+\frac{\varepsilon^{2}}{w_{*}-\varepsilon}\sum_{j\in
F^{c}}|x_{j}|$

Therefore, with $r=r_{F,\varepsilon}(\bm{x})$, we have

$\bm{w}^{\top}\bm{x}=r+\bm{w}_{F^{c}}^{\top}\,\bm{x}_{F^{c}}+\varepsilon\sum_{j=1}^{d}|x_{j}|=r+\varepsilon\sum_{j\in
F}|x_{j}|-\sum_{j\in F^{c}}(w_{j}-\varepsilon)|x_{j}|\\\\[7.0pt] \leq
r+\frac{\varepsilon
G}{w_{*}-\varepsilon}-\left(w_{*}-\frac{\varepsilon^{2}}{w_{*}-\varepsilon}-\varepsilon\right)\sum_{j\in
F^{c}}|x_{j}|\leq r+\frac{\varepsilon G}{w_{*}-\varepsilon},$

provided that $\varepsilon$ is small enough. Bounding the probability under
the integral by $1$ and changing the variables, we obtain

$\int_{\Omega_{F,-}}e^{\bm{w}^{\top}\bm{x}}\,\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{\chi}_{\bm{x}}(\bm{t})>\bm{x}\right\\}\mathop{d\bm{x}}\leq\int_{\Omega_{F,-}}e^{\bm{w}^{\top}\bm{x}}\mathop{d\bm{x}}=\int_{-\infty}^{G}\mathop{dr}\int\mathop{dS}e^{\bm{w}^{\top}\bm{x}}\,r^{d-1}\leq\\\\[7.0pt]
\leq\int_{-\infty}^{G}\int\mathop{dS}e^{r+\varepsilon
G/(w_{*}-\varepsilon)}r^{d-1}\mathop{dr}\mathop{dS}\leq c_{1}e^{\varepsilon
G/(w_{*}-\varepsilon)}\int_{-\infty}^{G}e^{(1+\varepsilon)r}\mathop{dr}=c_{1}\,e^{c_{2}G}.$

Next, we concentrate on the intergral over $\Omega_{F,+}$. By Piterbarg
inequality, we have the following uniform in $\bm{x}\in\Omega_{F,+}$ upper
bound:

$\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\eta_{\bm{x},F}(\bm{t})>\bm{x}\right\\}\leq
c_{3}\left(\frac{r(\bm{x})-G}{\sigma}\right)^{2/\gamma}\exp\left(-\frac{1}{2}\left(\frac{r(\bm{x})-G}{\sigma}\right)^{2}\right).$

Plugging this bound into the integral and changing the variables, we obtain

$\int_{\Omega_{F,+}}e^{\bm{w}^{\top}\bm{x}}\,\mathbb{P}\left\\{\exists\,\bm{t}\in[\bm{0},\bm{\Lambda}]\colon\bm{\chi}_{\bm{x}}(\bm{t})>\bm{x}\right\\}\mathop{d\bm{x}}\leq\\\\[7.0pt]
\leq
c_{3}\int_{\Omega_{F,+}}e^{\bm{w}^{\top}\bm{x}}\left(\frac{r(\bm{x})-G}{\sigma}\right)^{2/\gamma}\exp\left(-\frac{1}{2}\left(\frac{r(\bm{x})-G}{\sigma}\right)^{2}\right)\mathop{d\bm{x}}=\\\\[7.0pt]
=c_{3}\int_{G}^{\infty}\mathop{dr}\int\mathop{dS}e^{\bm{w}^{\top}\bm{x}}\left(\frac{r-G}{\sigma}\right)^{2/\gamma+d-1}\exp\left(-\frac{1}{2}\left(\frac{r-G}{\sigma}\right)^{2}\right).$

Note that with $w^{*}=\max_{i=1,\dots,d}w_{i}$ we have

$r=\sum_{i\in F}(w_{i}-\varepsilon)|x_{i}|-\varepsilon\sum_{i\in
F^{c}}|x_{i}|\geq\left(w^{*}-\varepsilon\right)\sum_{i\in
F}|x_{i}|-\varepsilon\sum_{i\in F^{c}}|x_{i}|$

and it follows that for all $\varepsilon<w^{*}$ the following bound holds:

$\varepsilon\sum_{i\in F}|x_{i}|\leq\frac{\varepsilon
r}{w^{*}-\varepsilon}+\frac{\varepsilon^{2}}{w^{*}-\varepsilon}\sum_{i\in
F^{c}}|x_{i}|.$

This bound yields

$\left(\bm{w},\bm{x}\right)=r+\varepsilon\sum_{i\in F}^{d}|x_{i}|-\sum_{i\in
F^{c}}(w_{i}-\varepsilon)|x_{i}|\leq\left(1+\frac{\varepsilon}{w^{*}-\varepsilon}\right)r-\left(w_{*}-\varepsilon-\frac{\varepsilon^{2}}{w^{*}-\varepsilon}\right)\sum_{i\in
F^{c}}|x_{i}|,$

from which for small enough $\varepsilon$ follows that
$\left(\bm{w},\bm{x}\right)\leq(1+\varepsilon^{\prime})r,$ with
$\varepsilon^{\prime}=\varepsilon/(w^{*}-\varepsilon)$. Hence,

$c_{3}\int_{G}^{\infty}\mathop{dr}\int\mathop{dS}e^{\bm{w}^{\top}\bm{x}}\left(\frac{r-G}{\sigma}\right)^{2/\gamma+d-1}\exp\left(-\frac{1}{2}\left(\frac{r-G}{\sigma}\right)^{2}\right)\leq\\\\[7.0pt]
\leq
c_{4}\int_{-\infty}^{\infty}e^{(1+\varepsilon^{\prime})r}\exp\left(-\frac{1}{2}\left(\frac{r-G}{\sigma}\right)^{2}\right)\mathop{dr}\leq
c_{5}\,e^{c_{6}(G+\sigma^{2})},$

where in the last step we used the Gaussian mgf formula
$\mathbb{E}\left\\{e^{t\mathcal{N}(\mu,\sigma^{2})}\right\\}=e^{t\mu+t^{2}\sigma^{2}/2}$
with $t=1+\varepsilon^{\prime}$. ∎

###  8.7. Double crossing: vicinity of the diagonal

###### Proof of Lemma 1.

We begin the proof with the following upper bound:

$\mathbb{P}\left\\{\exists\,\bm{t}\in D_{\varepsilon}\colon X(t_{1})>au,\
X(t_{2})<-bu\right\\}\leq\mathbb{P}\left\\{\exists\,\bm{t}\in
D_{\varepsilon}^{+}\colon X(t_{1})-X(t_{2})>(a+b)\,u\right\\}\\\
+\mathbb{P}\left\\{\exists\,\bm{t}\in D_{\varepsilon}^{-}\colon
X(t_{1})-X(t_{2})>(a+b)\,u\right\\},$

where

$D_{\varepsilon}^{+}=\left\\{\bm{t}=(t,s)\in[0,T]^{2}\colon t<s\leq
t+\varepsilon\right\\},\qquad
D_{\varepsilon}^{-}=\left\\{\bm{t}=(t,s)\in[0,T]^{2}\colon s<t\leq
s+\varepsilon\right\\}.$

Define a Gaussian field

$\mathcal{X}(s,l)\coloneqq
X(s+l)-X(s),\qquad(s,l)\in\mathbb{T}\coloneqq[0,T]\times[0,\varepsilon]$

and use it to coarsen the bound above:

$\mathbb{P}\left\\{\exists\,\bm{t}\in D_{\varepsilon}^{-}\colon
X(t_{1})-X(t_{2})>(a+b)\,u\right\\}\leq\mathbb{P}\left\\{\exists\,(s,l)\in\mathbb{T}\colon\mathcal{X}(s,l)>(a+b)\,u\right\\}.$

The variance of this Gaussian random field is

$\sigma^{2}(s,l)=\operatorname{Var}\\{\mathcal{X}(s,l)\\}=\mathbb{E}\left\\{\left[X(s+l)-X(s)\right]^{2}\right\\}\leq
f(\varepsilon).$

By Borell-TIS inequality (22), there exists $\mu>0$ such that for all $u>\mu$

$\mathbb{P}\left\\{\exists\,(s,l)\in\mathbb{T}\colon\mathcal{X}(s,l)>(a+b)\,u\right\\}\leq\exp\left(-\frac{(u-\mu)^{2}}{2\,f(\varepsilon)}\right).$

Since $f(\varepsilon)\to 0$ by the hypotheses of the theorem, for any
$\delta>0$ there exists some $\varepsilon>0$ such that
$f(\varepsilon)<1/4\,\delta$. Therefore,

$\mathbb{P}\left\\{\exists\,\bm{t}\in
D_{\varepsilon}\colon\mathcal{X}(s,l)>(a+b)\,u\right\\}\leq o\left(e^{-\delta
u^{2}}\right).$

∎

###  8.8. Double crossing for stationary processes: expansions

###### Proof of Lemma 2.

Consider $\bm{X}_{1}(\bm{t})=(X(t_{1}),-X(T-t_{2}))^{\top}$. We want to find
the expansion of

$R(\bm{t},\bm{s})=\mathbb{E}\left\\{\bm{X}_{1}(\bm{t})\,\bm{X}_{1}(\bm{s})^{\top}\right\\}=\begin{pmatrix}\rho(|t_{1}-s_{1}|)&-\rho(|T-s_{2}-t_{1}|)\\\
-\rho(|T-t_{2}-s_{1}|)&\rho(|t_{2}-s_{2}|)\end{pmatrix}$

near $\bm{t}=\bm{0}$. We have:

$\Sigma=\begin{pmatrix}1&-\rho(T)\\\ -\rho(T)&1\end{pmatrix},$

and

$\displaystyle\Sigma-R(\bm{t},\bm{s})$
$\displaystyle=\begin{pmatrix}1-\rho(|t_{1}-s_{1}|)&\rho(|T-s_{2}-t_{1}|)-\rho(T)\\\
\rho(|T-t_{2}-s_{1}|)-\rho(T)&1-\rho(|t_{2}-s_{2}|)\end{pmatrix}$
$\displaystyle=A_{2,1}\,t_{1}+A_{2,2}\,t_{2}+A_{2,1}^{\top}\,s_{2}+A_{2,2}^{\top}\,s_{2}+A_{5,1}\,|t_{1}-s_{1}|^{\alpha}+A_{5,2}\,|t_{2}-s_{2}|^{\alpha}$
$\displaystyle\quad+o\left(t_{1}+t_{2}+s_{1}+s_{2}+|t_{1}-s_{1}|^{\alpha}+|t_{2}-s_{2}|^{\alpha}\right),$

where the matrix coefficients are given by

$A_{2,1}=A_{2,2}^{\top}=-\rho^{\prime}(T)\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\quad A_{5,1}=\vartheta\begin{pmatrix}1&0\\\
0&0\end{pmatrix},\quad A_{5,2}=\vartheta\begin{pmatrix}0&0\\\
0&1\end{pmatrix}.$

Clearly, $\alpha_{1}=\alpha_{2}=\alpha$. Next, we need the optimal vector

$\bm{w}=\Sigma^{-1}(\bm{0})\,(a,b)^{\top}=\frac{1}{1-\rho^{2}(T)}\begin{pmatrix}1&\rho(T)\\\
\rho(T)&1\end{pmatrix}\begin{pmatrix}a\\\
b\end{pmatrix}=\frac{1}{1-\rho^{2}(T)}\begin{pmatrix}a+b\rho(T)\\\
b+a\rho(T)\end{pmatrix}$

to check whether we have correctly identified $\beta_{1}=\beta_{2}=1$. That
is, we need to check A2.4

$\xi_{1}=\bm{w}^{\top}\,A_{2,1}\,\bm{w}=\xi_{2}=\bm{w}^{\top}\,A_{2,2}\,\bm{w}=\frac{-\rho^{\prime}(T)(a+b\rho(T))(b+a\rho(T))}{(1-\rho^{2}(T))^{2}}>0.$

We also need to check A2.5:

$\varkappa_{1}=\bm{w}^{\top}\,A_{5,1}\,\bm{w}=\varkappa_{2}=\bm{w}^{\top}\,A_{5,2}\,\bm{w}=\frac{C(b+a\rho(T))^{2}}{(1-\rho^{2}(T))^{2}}>0.$

Assumption A3:  may be shown as follows:

$\displaystyle\mathbb{E}\left\\{\left|\bm{X}(\bm{t})-\bm{X}(\bm{s})\right|^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{(X(t_{1})-X(s_{1}))^{2}+(X(t_{2})-X(s_{2}))^{2}\right\\}$
$\displaystyle=2(1-\rho(|t_{1}-s_{1}|))+2(1-\rho(|t_{2}-s_{2}|))$
$\displaystyle\leq
c_{1}\Big{(}|t_{1}-t_{2}|^{\alpha}+|t_{2}-s_{2}|^{\alpha}\Big{)}$

with some constant $c_{1}>0$. The last inequality follows from the asymptotics
of $\rho(t)$ near $t=0$. Hence, Assumption A3:  is satisfied with
$\bm{\gamma}=(\alpha,\alpha)^{\top}$. ∎

###  8.9. Double crossing for fBm: minimization of generalized variance

###### Proof of Lemma 3.

We begin the proof by making use of the positive homogenity of the generalized
variance $\sigma_{a,b}^{-2}$. Namely, for $c>0$ we have

$\sigma_{a,b}^{-2}(c\bm{t})=c^{-2H}\sigma_{a,b}^{-2}(\bm{t}).$

Therefore, if we assume that $\bm{t}=(t_{1},t_{2})^{\top}$ lies in the lower
triangle $t_{1}>t_{2}$, we obtain from the equation above

$\sigma_{a,b}^{-2}(t_{1},t_{2})=\left(\frac{t_{1}}{T}\right)^{-2H}\sigma_{a,b}^{-2}\left(T,\frac{Tt_{2}}{t_{1}}\right)\geq\sigma_{a,b}^{-2}\left(T,\frac{Tt_{2}}{t_{1}}\right),$

since $(t_{1}/T)^{-2H}\geq 1$. Since $t_{1}>t_{2}$, we have that
$t_{2}^{\prime}=Tt_{2}/t_{1}\in(0,T)$ and it follows that

$\min_{t_{1}>t_{2}}\sigma_{a,b}^{-2}(t_{1},t_{2})\geq\min_{t_{2}^{\prime}\in[0,T]}\sigma_{a,b}^{-2}(T,t_{2}^{\prime}),$

hence, to minimize $\bm{t}\mapsto\sigma_{a,b}^{-2}(\bm{t})$ in the lower
triangle we only need to minimize $t_{2}\mapsto\sigma_{a,b}^{-2}(T,t_{2})$ in
$t_{2}\in(0,T)$. Similarly, to minimize
$\bm{t}\mapsto\sigma_{a,b}^{-2}(\bm{t})$ in the upper triangle, we only need
to minimize $t_{1}\mapsto\sigma_{a,b}^{-2}(t_{1},T)$ in $t_{1}\in(0,T)$.

If $a<b$, the following trivial inequality

$\sigma_{a,b}^{-2}(t,T)=\frac{a^{2}t^{2H}+2abr(t,T)+b^{2}T^{2H}}{(tT)^{2H}-r^{2}(t,T)}>\frac{a^{2}T^{2H}+2abr(t,T)+b^{2}t^{2H}}{(tT)^{2H}-r^{2}(t,T)}=\sigma_{a,b}^{-2}(T,t)$

shows, that the minimum over lower triangle is strictly smaller than the
minimum over upper triangle. Therefore, the global minimum lies in
$t_{1}>t_{2}$. Similarly, if $a>b$, the global minimum lies in $t_{1}<t_{2}$.
If $a=b$, we have two global minima.

Let us proceed to showing that the function

$t\mapsto\sigma_{a,b}^{-2}(T,s)=\frac{a^{2}s^{2H}+2abr(s,T)+b^{2}T^{2H}}{(sT)^{2H}-r^{2}(s,T)}$

possesses a unique minimum in $t\in(0,T)$. Without loss of generality, we may
rewrite this function as

(121)
$\sigma_{a,b}^{-2}(T,s)=\frac{a^{2}}{T^{2H}}\,D_{b/a}\left(\frac{s}{T}\right),$

where

$D_{\alpha}(s)\coloneqq\frac{\alpha^{2}+2\alpha
f(s)+s^{2H}}{s^{2H}-f^{2}(s)}=\frac{(\alpha+f(s))^{2}}{s^{2H}-f^{2}(s)}+1\qquad
s\in(0,1),$

where we introduced the function

$f(s)=\frac{1}{2}\Big{(}s^{2H}+1-(1-s)^{2H}\Big{)}.$

A straightforward approach would be to show that $D_{\alpha}^{\prime}(0+)<0$,
$D_{\alpha}^{\prime}(1-)>0$ and that $D_{\alpha}^{\prime\prime}>0$. The first
two claims are easily seen to be true, but, unfortunately, the third is false.
The idea we shall employ to get around this issue is to multiply the function
$D_{\alpha}^{\prime}$ by an appropriately chosen and strictly positive
function $U>0$, so that the roots of $D_{\alpha}^{\prime}U$ remained the same
as the roots of $D_{\alpha}^{\prime}$, but $D_{\alpha}^{\prime}U$ became
strictly increasing. We now proceed to finding such multiplier.

First, rewrite $D_{\alpha}^{\prime}$ collecting $\alpha$-free and
$\alpha$-dependent terms:

(122)
$D_{\alpha}^{\prime}(s)=\frac{2H(\alpha+f(s))}{(s^{2H}-f^{2}(s))^{2}}\Big{(}\alpha
G_{\alpha}(s)+G_{0}(s)\Big{)},$

where

$G_{\alpha}(s)=\frac{f(s)f^{\prime}(s)}{H}-s^{2H-1},\qquad
G_{0}(s)=\frac{f^{\prime}(s)s^{2H}}{H}-f(s)s^{2H-1}.$

We can drop the positive factor

$\frac{2H(\alpha+f(s))}{(s^{2H}-f^{2}(s))^{2}}>0,$

since the roots of $D_{\alpha}^{\prime}$ are that of
$\alpha\,G_{\alpha}(s)+G_{0}(s)$. Unfortunately, this remainder is still non-
monotone.

Proceding with the computations, we expand the derivatives and find that

$\displaystyle G_{\alpha}(s)$
$\displaystyle=f(s)\Big{(}s^{2H-1}+(1-s)^{2H-1}\Big{)}-s^{2H-1}$
$\displaystyle=f(s)(1-s)^{2H-1}-(1-f(s))s^{2H-1}$
$\displaystyle=f(s)(1-s)^{2H-1}-f(1-s)s^{2H-1}$
$\displaystyle=s^{2H-1}(1-s)^{2H-1}\Big{(}f(s)s^{1-2H}-f(1-s)(1-s)^{1-2H}\Big{)}.$

where in the second to last equality we used the identity $f(s)+f(1-s)=1$. We
can now represent $G_{\alpha}(s)$ as

$G_{\alpha}(s)=s^{2H-1}(1-s)^{2H-1}\Big{(}A(s)-A(1-s)\Big{)},\qquad
A(s)=f(s)s^{1-2H}.$

Similarly, but using the identity $f(s)-f(1-s)=s^{2H}-(1-s)^{2H}$, we obtain a
representation of $G_{0}(s)$

$\displaystyle G_{0}(s)$
$\displaystyle=\Big{(}s^{2H-1}+(1-s)^{2H-1}\Big{)}s^{2H}-f(s)s^{2H-1}$
$\displaystyle=\Big{(}s^{2H-1}+(1-s)^{2H-1}\Big{)}s^{2H}-\Big{(}f(1-s)+s^{2H}-(1-s)^{2H}\Big{)}s^{2H-1}$
$\displaystyle=-f(1-s)s^{2H-1}+\Big{(}(1-s)^{2H-1}s^{2H}+(1-s)^{2H}s^{2H-1}\Big{)}$
$\displaystyle=s^{2H-1}(1-s)^{2H-1}\Big{(}-A(1-s)+1\Big{)}.$

We can now rewrite (122) as

(123)
$D_{\alpha}^{\prime}(s)=\widetilde{D}_{\alpha}(s)\,\widetilde{G}_{\alpha}(s),\qquad$

with

(124) $\displaystyle\widetilde{G}_{\alpha}(s)$
$\displaystyle\coloneqq\alpha\Big{(}A(s)-A(1-s)\Big{)}-A(1-s)+1,$ (125)
$\displaystyle\widetilde{D}_{\alpha}(s)$
$\displaystyle\coloneqq\frac{2H(\alpha+f(s))s^{2H-1}(1-s)^{2H-1}}{(s^{2H}-f^{2}(s))^{2}}.$

We claim now that the function $\widetilde{G}_{\alpha}(s)$ is increasing.
Provided that this is true, we immediately obtain both existence and
uniqueness of the optimal point $s_{*}$, as well as positivity of the second
derivative at this point. Indeed,

(126)
$D_{\alpha}^{\prime\prime}(s_{*})=\widetilde{D}_{\alpha}^{\prime}(s_{*})\,\underbracket{\widetilde{G}_{\alpha}(s_{*})}_{=0}+\underbracket{\widetilde{D}_{\alpha}(s_{*})}_{>0}\,\underbracket{\widetilde{G}_{\alpha}^{\prime}(s_{*})}_{>0}>0.$

To prove this claim it clearly suffices to show that $A(s)$ is increasing.

We have,

$A^{\prime}(s)=f^{\prime}(s)s^{1-2H}+(1-2H)f(s)s^{-2H}$

and its positivity is equivalent to that of

$sf^{\prime}(s)+(1-2H)f(s).$

In case $H\leq 1/2$, the inequality

$sf^{\prime}(s)+(1-2H)f(s)>0$

is clear, since $f^{\prime}(s)>0$. If $H>1/2$, we use the Bernoulli inequality

$(1-s)^{2H-1}\leq 1-(2H-1)s,$

which gives

$\displaystyle sf^{\prime}(s)+(1-2H)f(s)$
$\displaystyle=(1-2H)+s^{2H}-(1-s)^{2H}+2H(1-s)^{2H-1}$
$\displaystyle\geq(1-2H)+s^{2H}-(1-s)\Big{(}1-(2H-1)s\Big{)}+2H(1-s)^{2H-1}$
$\displaystyle=s^{2H}-(2H-1)s^{2}+2H\Big{(}(1-s)^{2H-1}-(1-s)\Big{)}$
$\displaystyle\geq s^{2H}-(2H-1)s^{2}=s^{2}\Big{(}s^{2H-2}-2H+1\Big{)}$
$\displaystyle\geq 2s^{2}\Big{(}1-H\Big{)}>0.$

As a corollary of the above, we obtain
$\alpha\,G_{\alpha}(s_{*})+G_{0}(s_{*})=0$ or

(127)
$\alpha\Bigg{[}r(s_{*},1)\Big{(}s_{*}^{2H-1}+(1-s_{*})^{2H-1}\Big{)}-s_{*}^{2H-1}\Bigg{]}+\Bigg{[}s_{*}^{2H}\Big{(}s_{*}^{2H-1}+(1-s_{*})^{2H-1}\Big{)}-r(s_{*},1)s_{*}^{2H-1}\Bigg{]}=0,$

which will be useful for us in Lemma 4.

We have thus shown that the function

$\sigma_{\bm{b}}^{-2}(t,s)=\frac{a^{2}s^{2H}+2abr(t,s)+b^{2H}t^{2}}{(ts)^{2H}-r^{2}(t,s)}$

posesses a unique minimum in the lower triangle. By (121), we see that this
point is given by

$t_{*}=Ts_{*},$

where $s_{*}$ is the minimizer of $D$. Moreover, we have by (123)

$\frac{\partial\sigma_{\bm{b}}^{-2}}{\partial
s}(T,t_{*})=\frac{a^{2}}{T^{2H}}\widetilde{D}_{b/a}(s_{*})\underbracket{\widetilde{G}_{b/a}(s_{*})}_{=0}=0$

and by (126) we obtain

$\kappa_{2}\coloneqq\frac{\partial^{2}\sigma_{\bm{b}}^{-2}}{\partial
s^{2}}(T,t_{*})=\frac{a^{2}}{T^{2H}}\,D_{b/a}^{\prime\prime}(s_{*})=\frac{a^{2}}{T^{2H}}\,\widetilde{D}_{b/a}(s_{*})\widetilde{G}_{b/a}^{\prime}(s_{*})>0$

Similarly to (121), let us rewrite $\sigma_{\bm{b}}^{-2}$ as follows:

$\sigma_{\bm{b}}^{-2}(t,s)=\frac{a^{2}}{t^{2H}}\,D_{b/a}\left(\frac{s}{t}\right).$

Therefore,

$-\kappa_{1}\coloneqq\frac{\partial\sigma_{\bm{b}}^{-2}}{\partial
t}(T,t_{*})=-\frac{2Ha^{2}}{T^{2H+1}}\,D_{b/a}(s_{*})+\frac{a^{2}}{T^{2H}}\,\left[\frac{-s_{*}}{T^{2}}\right]\underbracket{D_{b/a}^{\prime}(s_{*})}_{=0}<0.$

Finally, we have

$\sigma_{\bm{b}}^{-2}(\bm{t}_{*})-\sigma_{\bm{b}}^{-2}(\bm{t}_{*}-\bm{\tau})\sim-\kappa_{1}\,\tau_{1}-\kappa_{2}\,\tau_{2}^{2}.$

∎

###  8.10. Double crossing for fBm: matrix expansions

###### Proof of Lemma 4.

Let $\bm{t}_{*}=(T,t_{*})$ be a point in $[0,T]^{2}$ minimizing the
generalized variance $\sigma_{a,b}^{-2}(\bm{t})$.

Recall that

$\Sigma(\bm{t}_{*})=\begin{pmatrix}T^{2H}&-r(T,t_{*})\\\
-r(T,t_{*})&t_{*}^{2H}\end{pmatrix},\qquad
R(\bm{t},\bm{s})=\begin{pmatrix}r(t_{1},s_{1})&-r(t_{1},s_{2})\\\
-r(t_{2},s_{1})&r(t_{2},s_{2})\end{pmatrix},$

where

$r(t,s)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right).$

We have

$\Sigma(\bm{t}_{*})-R(\bm{t}_{*}+\,\bm{t},\bm{t}_{*}+\bm{s})=\begin{pmatrix}T^{2H}-r(T+t_{1},T+s_{1})&-r(T,t_{*})+r(T+t_{1},t_{*}+s_{2})\\\\[7.0pt]
-r(t_{*},T)+r(t_{*}+t_{2},T+s_{1})&t_{*}^{2H}-r(t_{*}+t_{2},t_{*}+s_{2})\end{pmatrix}.$

For the top left cell, we have:

$T^{2H}-r(T+t_{1},T+s_{1})=T^{2H}-\frac{1}{2}\left((T+t_{1})^{2H}+(T+s_{1})^{2H}-|t_{1}-s_{1}|^{2H}\right).$

Here is the expression for the top right cell:

$r(T+t_{1},t_{*}+s_{2})-r(T,t_{*})=\frac{1}{2}\left((T+t_{1})^{2H}-T^{2H}\right)+\frac{1}{2}\left((t_{*}+s_{2})^{2H}-t_{*}^{2H}\right)\\\\[7.0pt]
-\frac{1}{2}\left(|T-t_{*}+t_{1}-s_{2}|^{2H}-|T-t_{*}|^{2H}\right)$

and similarly for the remaining two. Let us compute the first order
coefficients of different contributions. Jumping ahead, we will be giving
these coefficients names corresponding to their roles within Assumption A2: .
Recall that the first index $i$ in $A_{i,j}$ corresponds to the order (first
or second) of the contribution, while the second indicates the variable
$t_{j}$. The coefficients of the corresponding $\bm{s}$-terms can be expressed
as transpositions of these.

First, the only two terms which depend on the difference are the following:

$\left|t_{1}-s_{1}\right|^{2H}\colon\quad
A_{5,1}\coloneqq\frac{1}{2}\begin{pmatrix}1&0\\\
0&0\end{pmatrix},\qquad\left|t_{2}-s_{2}\right|^{2H}\colon\quad
A_{5,2}\coloneqq\frac{1}{2}\begin{pmatrix}0&0\\\
0&1\end{pmatrix},\qquad\alpha_{1}=\alpha_{2}=2H.$

Next, we proceed to the power-type contributions of the leading order:

$\displaystyle t_{1}\colon\quad A_{2,1}\coloneqq
H\begin{pmatrix}-T^{2H-1}&T^{2H-1}-|T-t_{*}|^{2H-1}\\\ 0&0\end{pmatrix},$
$\displaystyle t_{2}\colon\quad A_{1,2}\coloneqq H\begin{pmatrix}0&0\\\
t_{*}^{2H-1}+|T-t_{*}|^{2H-1}&-t_{*}^{2H-1}\end{pmatrix}.$

We will show below that

(128) $\bm{w}^{\top}A_{2,1}\,\bm{w}>0\quad\text{and}\quad
A_{1,2}\,\bm{w}(\bm{t})\sim 0,$

whence the names $A_{2,1}$ and $A_{1,2}$. It also explains why we do not need
to compute the second order in $t_{1}$. However, we do need to find the second
order in the second coordinate:

(129) $t_{2}^{2}\colon\quad A_{2,2}\coloneqq
H\left(H-\frac{1}{2}\right)\begin{pmatrix}0&0\\\
t_{*}^{2H-2}+|T-t_{*}|^{2H-2}&-t_{*}^{2H-2}\end{pmatrix},$

and the coefficient $A_{6,2,2}$ of $t_{2}\,s_{2}$ is zero. To show (128), we
need the inverse of $\Sigma$

$\Sigma^{-1}(\bm{t})=\frac{1}{t_{1}^{2H}t_{2}^{2H}-r^{2}(t_{1},t_{2})}\begin{pmatrix}t_{2}^{2H}&r(t_{1},t_{2})\\\
r(t_{1},t_{2})&t_{1}^{2H}\end{pmatrix}$

and its action on the vector $\bm{w}(\bm{t})$:

$\bm{w}(\bm{t})=\Sigma^{-1}(\bm{t})\,\bm{b}=\frac{1}{t_{1}^{2H}t_{2}^{2H}-r^{2}(t_{1},t_{2})}\begin{pmatrix}t_{2}^{2H}\,a+r(t_{1},t_{2})\,b\\\\[7.0pt]
r(t_{1},t_{2})\,a+t_{1}^{2H}\,b\end{pmatrix}.$

Using the following identity (127) from Lemma 3

$b\Bigg{[}r(s_{*},1)\Big{(}s_{*}^{2H-1}+(1-s_{*})^{2H-1}\Big{)}-s_{*}^{2H-1}\Bigg{]}+a\Bigg{[}s_{*}^{2H}\Big{(}s_{*}^{2H-1}+(1-s_{*})^{2H-1}\Big{)}-r(s_{*},1)s_{*}^{2H-1}\Bigg{]}=0$

we can show that Assumptions A2.3 to A2.5 are satisfied with

$A_{1,2}\,\bm{w}(\bm{t})\sim
0,\quad\bm{w}^{\top}A_{2,1}\,\bm{w}>0,\quad\bm{w}^{\top}\,A_{5,2}\,\bm{w}>0,\quad\beta_{1}=1,\quad\beta_{2}=2,\quad\mathcal{F}=\\{2\\},$

which gives

$\Sigma-R(\bm{t}_{*}+\bm{t},\bm{t}_{*}+\bm{s})=\Big{[}A_{2,1}\,t_{1}+A_{1,2}\,t_{2}+A_{2,2}\,t_{2}^{2}\Big{]}+\Big{[}A_{2,1}^{\top}\,s_{1}+A_{1,2}^{\top}\,s_{2}+A_{2,2}^{\top}\,s_{2}^{2}\Big{]}\\\\[7.0pt]
+A_{5,1}\,|t_{1}-s_{1}|^{2H}+A_{5,2}\,|t_{2}-s_{2}|^{2H}+o\left(\sum_{i=1}^{n}\Big{[}t_{1}+t_{2}^{2}+s_{1}+s_{2}^{2}\Big{]}\right),$

which verifies A2.1 and A2.2. Finally, $\mathcal{F}\times\mathcal{F}$ contains
one element $(2,2)$, and by (129) we have that

$A_{6,2,2}+A_{1,2}\Sigma^{-1}A_{1,2}^{\top}=A_{1,2}\Sigma^{-1}A_{1,2}=C_{2}\,C_{2}^{\top}\quad\text{with}\quad
C_{2}=A_{1,2}\,\Sigma^{-1/2},$

so A2.6 is satisfied. ∎

## Acknowledgements

The authors kindly acknowledge the financial support by SNSF Grant
200021-196888.

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