Asymptotic expansions of integral functionals of vector valued ε-correlated processes

J. vom Scheidt, H.-J. Starkloff, R. Wunderlich

Abstract

In the paper asymptotic expansions for second-order moments of integral functionals of a family of complex vector valued wide-sense stationary and ε-correlated random processes are considered. The asymptotic expansions are derived for ε → 0 and extend the results for real valued functionals to the complex vector valued case.

1 Introduction

In this paper asymptotic expansions for second-order moments of integral functionals of the type

εr(ω) = ∫_{D} Q(s)^{ε}f(s, ω) ds,  ε > 0,  (1)

are considered, where Q(·) is a deterministic complex n × n-matrix valued function on D ⊂ R, n ∈ N, and (^{ε}f(·)), ε > 0, denotes a family of complex vector valued wide-sense stationary and ε-correlated random processes. The given results extend the corresponding expansions for real valued processes in [7]. In the case examined here the noncommutativity of matrix multiplication has to be taken into account. In the paper vectors are assumed to be columns and ^{∗} denotes the adjoint of a vector or matrix.

Analogously to [7] we will suppose

Assumption 1

1. (^{ε}f(·)), ε > 0, are complex vector valued wide-sense stationary processes with matrix correlation functions

E{^{ε}f(s)^{ε}f^{∗}(t)} = ^{ε}R(t − s);

2. (^{ε}f(·)), ε > 0, are centered processes, i.e. E{^{ε}f(s)} = 0; 3. (^{ε}f(·)), ε > 0, are ε-correlated processes, i.e. ^{ε}R(s) = 0 for |s| ≥ ε;

4. the matrix correlation functions ^{ε}R(·), ε > 0, are generated by a matrix correlation function R(·) of a complex vector valued 1–correlated wide-sense stationary process, i.e.

εR(s) = R (\frac{s}{ε}) , s ∈ R, ε > 0;

5. the matrix correlation function R(·) is continuous on R, hence the processes (^{ε}f(·)), ε > 0, are continuous in mean square on R.

The integral in (1) is assumed to exist in mean square sense, under weak conditions it coincides a.s. with the pathwise integral. From condition 2 of Assumption 1 it follows that the random vectors ^{ε}r(·), ε > 0, are centered, i.e. E{^{ε}r} = 0. Also in the vector valued case correlation moments of the processes play a crucial role.

Definition 1 Let R(·) = (R_{ij}(·))_{i,j=1,...,n} be a continuous matrix correlation function of a complex vector valued wide-sense stationary process (not necessarily 1–correlated) and j ∈ N_{0} = {0, 1, 2, …} with

∫_{−∞}^{∞} |s|^{j}|R_{ij}(s)| ds < ∞, i, j = 1, …, n.

Then the matrix

μ_{j} = ∫_{−∞}^{∞} s^{j}R(s) ds

is called the correlation moment matrix of j-th order of the correlation function or the random process and the matrix

ν_{j} = ∫_{−∞}^{∞} |s|^{j}R(s) ds

is called the absolute correlation moment matrix of j-th order.

With the notation

μ_{j}^{+} = ν_{j}^{+} = ∫_{0}^{∞} s^{j}R(s) ds

we find from the relation R(s) = R^{∗}(−s)

μ_{j} = ν_{j}^{+} + (−1)^{j}[ν_{j}^{+}]^{∗},
ν_{j} = ν_{j}^{+} + [ν_{j}^{+}]^{∗},

hence it also holds

μ_{j}^{∗} = (−1)^{j}μ_{j}, ν_{j}^{∗} = ν_{j}.

In the following treatment the function Q(·) satisfies Assumption 2 The deterministic complex matrix valued function Q(·) is N− times continuously differentiable on D, N ∈ N_{0}, Q^{(N)}(·) is absolutely continuous on D and the derivatives of Q(·) up to the order (N + 1) belong to the space L^{2}(D) ∩ L^{1}(D) of matrix valued functions.

For such functions the Taylor expansion formula with exact integral representation of the remainder (cf. e.g. [2], Sec. 5.4) remains valid, for the integration by parts formula for absolutely continuous function see e.g. [3], Ch. IX, §7.

2 Expansion of covariance matrices

2.1 The case D = R

For functionals of the type

εr = ∫_{−∞}^{∞} Q(s)^{ε}f(s) ds

we get

E{^{ε}r^{ε}r^{∗}} = ε ∫_{−∞}^{∞}∫_{−∞}^{∞} Q(t − εu)R(u)Q^{∗}(t) dt du
= ε ∫_{−1}^{1}∫_{−∞}^{∞} Q(t − εu)R(u)Q^{∗}(t) dt du.

From the Taylor expansion

Q(t − εu) = ∑_{j=0}^{N} Q^{(j)}(t)\frac{(−εu)^{j}}{j!} + ρ̃_{N+1}(t, u, ε)

with

ρ̃_{N+1}(t, u, ε) = \frac{1}{N!} ∫_{t}^{t−εu} Q^{(N+1)}(v)(t − εu − v)^{N} dv

we find

Theorem 1 Let (^{ε}f(s); s ∈ R), ε > 0, be a family of C^{n}-valued random processes satisfying Assumption 1 and Q(·) a matrix valued function satisfying Assumption 2 on D = R. Then it holds

E\{^{ε}r^{ε}r^{∗}\} = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} q_{j} + ρ_{N+1}(ε)

with

q_{j} = ∫_{−∞}^{∞} Q^{(j)}(t)μ_{j}^{∗}Q^{∗}(t) dt

and the remainder term

ρ_{N+1}(ε) = \frac{ε}{N!} ∫_{−1}^{1} ∫_{−∞}^{∞} ∫_{t}^{t−εu} Q^{(N+1)}(v)(t − εu − v)^{N}R(u)Q^{∗}(t) dv dt du.

Integration by parts with respect to the quantities q_{j} leads for k = 0, …, j to

q_{j} = (−1)^{k} ∫_{−∞}^{∞} Q^{(j−k)}(t)μ_{j}^{∗}[Q^{(k)}(t)]^{∗} dt,

especially

q_{2k} = (−1)^{k} ∫_{−∞}^{∞} Q^{(k)}(t)μ_{2k}^{∗}[Q^{(k)}(t)]^{∗} dt,  k = 0, …, [\frac{N}{2} ] ,
q_{2k+1} = (−1)^{k} ∫_{−∞}^{∞} Q^{(k+1)}(t)μ_{2k+1}^{∗}[Q^{(k)}(t)]^{∗} dt, k = 0, …, [\frac{N − 1}{2}] .

2.2 The case D = R_{+}

For functionals of the type

εr = ∫_{0}^{∞} Q(s)^{ε}f(s) ds

we get

E\{^{ε}r^{ε}r^{∗}\} = ∫_{0}^{∞} ∫_{0}^{∞} Q(s)R (\frac{t − s}{ε}) Q^{∗}(t) ds dt
= ε ∫_{−∞}^{0}∫_{0}^{∞} Q(t − εu)R(u)Q^{∗}(t) dt du
+ ε ∫_{0}^{∞}∫_{εu}^{∞} Q(t − εu)R(u)Q^{∗}(t) dt du
= ε ∫_{−∞}^{0}∫_{0}^{∞} Q(t − εu)R(u)Q^{∗}(t) dt du
+ ε ∫_{0}^{∞}∫_{0}^{∞} Q(t)R(u)Q^{∗}(t + εu) dt du
= ε ∫_{−1}^{0}∫_{0}^{∞} Q(t + ε|u|)R(u)Q^{∗}(t) dt du
+ ε ∫_{0}^{1}∫_{0}^{∞} Q(t)R(u)Q^{∗}(t + ε|u|) dt du
= [ε ∫_{0}^{1}∫_{0}^{∞} Q(t)R(u)Q^{∗}(t + εu) dt du]^{∗}
+ ε ∫_{0}^{1}∫_{0}^{∞} Q(t)R(u)Q^{∗}(t + εu) dt du.

Taylor expansion gives

Q^{∗}(t + εu) = ∑_{j=0}^{N}[Q^{(j)}(t)]^{∗} \frac{(εu)^{j}}{j!} + ρ̃_{N+1}(t, u, ε)

with

ρ̃_{N+1}(t, u, ε) = \frac{1}{N!} ∫_{t}^{t+εu}[Q^{(N+1)}(v)]^{∗}(t + εu − v)^{N} dv.

Theorem 2 Let (^{ε}f(s); s ∈ R_{+}), ε > 0, be a family of C^{n}-valued random processes satisfying Assumption 1 and Q(·) a matrix valued function satisfying Assumption 2 on D = R_{+}. Then it holds

E\{^{ε}r^{ε}r^{∗}\} = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{q_{j} + q_{j}^{∗}\} + ρ_{N+1}(ε)

with

q_{j} = ∫_{0}^{∞} Q(t)ν_{j}^{+}[Q^{(j)}(t)]^{∗} dt

and the remainder term

ρ_{N+1}(ε) = \frac{ε}{N!} \{∫_{0}^{1}∫_{0}^{∞}∫_{t}^{t+εu} Q(t)(t + εu − v)^{N}R(u)[Q^{(N+1)}(v)]^{∗}dv dtdu

+ [∫_{0}^{1}∫_{0}^{∞}∫_{t}^{t+εu} Q(t)(t + εu − v)^{N}R(u)[Q^{(N+1)}(v)]^{∗}dv dtdu]^{∗}} .

Integration by parts with respect to the quantities q_{j} leads for k = 1, …, j to

q_{j} = ∑_{l=0}^{k−1}(−1)^{l+1}Q^{(l)}(0)ν_{j}^{+}[Q^{(j−1−l)}(0)]^{∗} + (−1)^{k}∫_{0}^{∞}Q^{(k)}(t)ν_{j}^{+}[Q^{(j−k)}(t)]^{∗} dt,

especially

q_{2k} = ∑_{l=0}^{k−1}(−1)^{l+1}Q^{(l)}(0)ν_{2k}^{+}[Q^{(2k−1−l)}(0)]^{∗}

+ (−1)^{k}∫_{0}^{∞}Q^{(k)}(t)ν_{2k}^{+}[Q^{(k)}(t)]^{∗} dt, k = 0, …, [\frac{N}{2} ] ,

q_{2k+1} = ∑_{l=0}^{k−1}(−1)^{l+1}Q^{(l)}(0)ν_{2k+1}^{+}[Q^{(2k−l)}(0)]^{∗} +

+ (−1)^{k}∫_{0}^{∞}Q^{(k)}(t)ν_{2k+1}^{+}[Q^{(k+1)}(t)]^{∗} dt, k = 0, …, [\frac{N − 1}{2}] .

2.3 The case D = [a, b]

For functionals of the type

εr = ∫_{a}^{b} Q(s)^{ε}f(s) ds

with finite values a, b ∈ R, b − a ≥ ε, we get

E\{^{ε}r^{ε}r^{∗}\} = ∫_{a}^{b} ∫_{a}^{b} Q(s)R (\frac{t − s}{ε}) Q^{∗}(t) ds dt
= ε ∫ 0_{\frac{a−b}{ε}} ∫_{a}^{b+εu} Q(t − εu)R(u)Q^{∗}(t) dt du +

+ ε ∫0^{\frac{b−a}{ε}} ∫_{a+εu}^{b} Q(t − εu)R(u)Q^{∗}(t) dt du

= ε ∫_{−1}^{0}∫_{a}^{b−ε|u|} Q(t + ε|u|)R(u)Q^{∗}(t) dt du +

+ ε ∫_{0}^{1} ∫_{a}^{b−ε|u|} Q(t)R(u)Q^{∗}(t + ε|u|) dt du

= [ε ∫_{0}^{1} ∫_{a}^{b−εu} Q(t)R(u)Q^{∗}(t + εu) dt du]^{∗}

+ ε ∫_{0}^{1} ∫_{a}^{b−εu} Q(t)R(u)Q^{∗}(t + εu) dt du.

Defining for ε ≥ 0, 0 ≤ u ≤ 1 the matrix valued function

ϕ(ε, u) := ∫_{a}^{b−εu} Q(t)R(u)Q^{∗}(t + εu) dt

we find from Assumption 2

ϕ(ε, u) = ∑_{j=0}^{N} \frac{ε^{j}}{j!} \frac{∂^{j}ϕ(0, u)}{∂ε^{j}} + \frac{1}{N!} ∫_{0}^{ε} \frac{∂^{N+1}ϕ(v, u)}{∂v^{N+1}}(ε − v)^{N} dv.

Calculation of the matrices \frac{∂^{j}ϕ(0,u)}{∂ε^{j}}, 0 ≤ u ≤ 1, gives

ϕ(ε, u) = ∫_{a}^{b−εu} Q(t)R(u)Q^{∗}(t + εu) dt,
ϕ(0, u) = ∫_{a}^{b} Q(t)R(u)Q^{∗}(t) dt,
\frac{∂ϕ(ε, u)}{∂ε} = ∫_{a}^{b−εu} Q(t)uR(u)[Q′(t + εu)]^{∗} dt − Q(b − εu)uR(u)Q^{∗}(b),
\frac{∂ϕ(0, u)}{∂ε} = ∫_{a}^{b} Q(t)uR(u)[Q′(t)]^{∗} dt − Q(b)uR(u)Q^{∗}(b),
\frac{∂^{2}ϕ(ε, u)}{∂ε^{2}} = ∫_{a}^{b−εu} Q(t)u^{2}R(u)[Q′′(t + εu)]^{∗} dt − Q(b − εu)u^{2}R(u)[Q′(b)]^{∗}
+Q′(b − εu)u^{2}R(u)Q^{∗}(b),
\frac{∂^{2}ϕ(0, u)}{∂ε^{2}} = ∫_{a}^{b} Q(t)u^{2}R(u)[Q′′(t)]^{∗} dt − Q(b)u^{2}R(u)[Q′(b)]^{∗}
+Q′(b)u^{2}R(u)Q^{∗}(b),

in general for j = 0, 1, …, N + 1

\frac{∂^{j}ϕ(ε, u)}{∂ε^{j}} = ∫_{a}^{b−εu} Q(t)u^{j}R(u)[Q^{(j)}(t + εu)]^{∗} dt

− ∑_{l=0}^{j−1}(−1)^{l}Q^{(l)}(b − εu)u^{j}R(u)[Q^{(j−1−l)}(b)]^{∗},

\frac{∂^{j}ϕ(0, u)}{∂ε^{j}} = ∫_{a}^{b} Q(t)u^{j}R(u)[Q^{(j)}(t)]^{∗} dt

− ∑_{l=0}^{j−1}(−1)^{l}Q^{(l)}(b)u^{j}R(u)[Q^{(j−1−l)}(b)]^{∗}.

Theorem 3 Let (^{ε}f(s); s ∈ [a, b]), ε > 0, be a family of C^{n}-valued random processes satisfying Assumption 1 and Q(·) a matrix valued function satisfying Assumption 2 on D = [a, b]. Then it holds

E\{^{ε}r^{ε}r^{∗}\} = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{q_{j} + q_{j}^{∗}\} + ρ_{N+1}(ε)

with

q_{j} = ∫_{a}^{b} Q(t)ν_{j}^{+}[Q^{(j)}(t)]^{∗} dt − ∑_{l=0}^{j−1}(−1)^{l}Q^{(l)}(b)ν_{j}^{+}[Q^{(j−1−l)}(b)]^{∗}

and the remainder term

ρ_{N+1}(ε) = \frac{ε}{N!} \{∫_{0}^{1} ∫_{0}^{ε} \frac{∂^{N+1}ϕ(v, u)}{∂v^{N+1}}(ε − v)^{N} dv du

+ [∫_{0}^{1} ∫_{0}^{ε} \frac{∂^{N+1}ϕ(v, u)}{∂v^{N+1}}(ε − v)^{N} dv du]^{∗}\} .

3 Expansion of matrix correlation functions

Considering the complex vector valued wide-sense stationary processes

εg(t) = ∫_{−∞}^{t} Q(t − s)^{ε}f(s) ds = ∫_{0}^{∞} Q(u)^{ε}f(t − u) du, ε > 0,

its matrix correlation functions can be written as

εR_{gg}(τ) = E{^{ε}g(t)^{ε}g^{∗}(t + τ)}
= ∫_{−∞}^{t} ∫_{−∞}^{t+τ} Q(t − s_{1})E{^{ε}f(s_{1})^{ε}f^{∗}(s_{2})}Q^{∗}(t + τ − s_{2}) ds_{2} ds_{1}
= ∫_{0}^{∞} ∫_{0}^{∞} Q(u_{1})^{ε}R_{ff}(τ + u_{1} − u_{2})Q^{∗}(u_{2}) du_{2} du_{1}

and it can be obtained

εR_{gg}(τ) = ∫_{−∞}^{0} ∫_{−v_{1}}^{∞} Q(v_{1} + v_{2})^{ε}R_{ff}(τ + v_{1})Q^{∗}(v_{2}) dv_{2} dv_{1} +

+ ∫_{0}^{∞} ∫_{0}^{∞} Q(v_{1} + v_{2})^{ε}R_{ff}(τ + v_{1})Q^{∗}(v_{2}) dv_{2} dv_{1}

= ∫_{−∞}^{0} ∫_{0}^{∞} Q(v_{2})^{ε}R_{ff}(τ + v_{1})Q^{∗}(v_{2} − v_{1}) dv_{2} dv_{1} +

+ ∫_{0}^{∞} ∫_{0}^{∞} Q(v_{1} + v_{2})^{ε}R_{ff}(τ + v_{1})Q^{∗}(v_{2}) dv_{2} dv_{1}

= ∫_{−∞}^{∞} ∫_{0}^{∞} Q(v_{2} + v_{1}^{+})^{ε}R_{ff}(τ + v_{1})Q^{∗}(v_{2} + v_{1}^{−}) dv_{2} dv_{1}
= ∫_{−∞}^{∞} ∫_{0}^{∞} Q(u + (v − τ)^{+})^{ε}R_{ff}(v)Q^{∗}(u + (v − τ)^{−}) du dv
= ∫_{−∞}^{∞} ∫_{0}^{∞} Q(u + (v − τ)^{+})R (\frac{v}{ε} ) Q^{∗}(u + (v − τ)^{−}) du dv
= ε ∫_{−1}^{1} ∫_{0}^{∞} Q(u + (εv − τ)^{+})R(v)Q^{∗}(u + (εv − τ)^{−}) du dv.

Now, Taylor expansion of the matrix function Q(·) in the neighbourhoods of the points u + |τ| is considered. In the case τ = 0 it yields

εR_{gg}(0) = ε ∫_{−1}^{0} ∫_{0}^{∞} Q(u)R(v)Q^{∗}(u + |εv|) du dv
+ ε ∫_{0}^{1} ∫_{0}^{∞} Q(u + |εv|)R(v)Q^{∗}(u) du dv
= + [ε ∫_{0}^{1} ∫_{0}^{∞} Q(u + εv)R(v)Q^{∗}(u) du dv]^{∗}
+ ε ∫_{0}^{1} ∫_{0}^{∞} Q(u + εv)R(v)Q^{∗}(u) du dv
= ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{[∫_{0}^{1} ∫_{0}^{∞} Q^{(j)}(u)v^{j}R(v)Q^{∗}(u) du dv]^{∗}
+ ∫_{0}^{1} ∫_{0}^{∞} Q^{(j)}(u)v^{j}R(v)Q^{∗}(u) du dv} + ρ_{N+1}(ε, 0)
= ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{q_{j} + q_{j}^{∗}\} + ρ_{N+1}(ε, 0)

with

q_{j} = ∫_{0}^{∞} Q^{(j)}(u)ν_{j}^{+}Q^{∗}(u) du, j = 0, …, N,

and the remainder term ρ_{N+1}(ε, 0). For τ > 0 we obtain

εR_{gg}(τ) = ε ∫_{−1}^{\frac{τ}{ε} ∧1} ∫_{0}^{∞} Q(u)R(v)Q^{∗}(u + τ − εv) du dv

+ ε ∫_{\frac{τ}{ε} ∧1}^{1} ∫_{0}^{∞} Q(u + εv − τ)R(v)Q^{∗}(u) du dv,

where a ∧ b := min(a, b). Taylor expansion in the neighbourhoods of the points u + τ results in

Q^{∗}(u + τ − εv) = ∑_{j=0}^{N}[Q^{(j)}(u + τ)]^{∗} \frac{(−εv)^{j}}{j!} + ρ̃_{N+1}^{1}(u, v, ε, τ),

Q(u + εv − τ) = ∑_{j=0}^{N} Q^{(j)}(u + τ)\frac{ε^{j}}{j!} (v − \frac{2τ}{ε} )^{j} + ρ̃_{N+1}^{2}(u, v, ε, τ),

hence

εR_{gg}(τ) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{∫_{−1}^{\frac{τ}{ε} ∧1} ∫_{0}^{∞} Q(u)(−v)^{j}R(v)[Q^{(j)}(u + τ)]^{∗} du dv

+ ∫_{\frac{τ}{ε} ∧1}^{1} ∫_{0}^{∞} Q^{(j)}(u + τ) (v − \frac{2τ}{ε} )^{j} R(v)Q^{∗}(u) du dv\} + ρ_{N+1}(ε, τ)

= ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{∫_{−1}^{1} ∫_{0}^{∞} Q(u)(−v)^{j}R(v)[Q^{(j)}(u + τ)]^{∗} du dv

− ∫_{\frac{τ}{ε} ∧1}^{1} ∫_{0}^{∞} Q(u)(−v)^{j}R(v)[Q^{(j)}(u + τ)]^{∗} du dv

+ ∫_{\frac{τ}{ε} ∧1}^{1} ∫_{0}^{∞} Q^{(j)}(u + τ) (v − \frac{2τ}{ε} )^{j} R(v)Q^{∗}(u) du dv\} + ρ_{N+1}(ε, τ).

Using the relation (−1)^{j}μ_{j} = μ_{j}^{∗} we conclude

Theorem 4 Let (^{ε}f(s); s ∈ R), ε > 0, be a family of C^{n}-valued random processes satisfying Assumption 1 and Q(·) a matrix valued function satisfying Assumption 2 on D = R_{+}. Then it holds for τ > 0

εR_{gg}(τ) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{q_{j}(τ) + 1_{\{0<τ<ε\}}(τ)c_{j}(τ)\} + ρ_{N+1}(ε, τ)

with

q_{j}(τ) = ∫_{0}^{∞} Q(u)μ_{j}^{∗}[Q^{(j)}(u + τ)]^{∗} du,  j = 0, …, N,
c_{j}(τ) = ∫_{0}^{∞} Q^{(j)}(u + τ) ∫ 1_{\frac{τ}{ε}} (v − \frac{2τ}{ε} )^{j} R(v) dv Q^{∗}(u) du

+ (−1)^{j+1} ∫_{0}^{∞} Q(u) ∫ 1_{\frac{τ}{ε}} v^{j}R(v) dv [Q^{(j)}(u + τ)]^{∗} du, j = 0, …, N,

ρ_{N+1}(ε, τ) = ε \{∫_{−1}^{\frac{τ}{ε} ∧1} ∫_{0}^{∞} Q(u)R(v)ρ̃_{N+1}^{1}(u, v, ε, τ) du dv
+ ∫_{\frac{τ}{ε} ∧1}^{1} ∫_{0}^{∞} ρ̃_{N+1}^{2}(u, v, ε, τ)R(v)Q^{∗}(u) du dv\} ,
ρ̃_{N+1}^{1}(u, v, ε, τ) = \frac{1}{N!} ∫_{u+τ}^{u+τ−εv}[Q^{(N+1)}(w)]^{∗}(u + τ − εv − w)^{N} dw,
ρ̃_{N+1}^{2}(u, v, ε, τ) = \frac{1}{N!} ∫_{u+τ}^{u+εv−τ} Q^{(N+1)}(w)(u + εv − τ − w)^{N} dw.

For τ = 0 it holds

εR_{gg}(0) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{q_{j} + q_{j}^{∗}\} + ρ_{N+1}(ε, 0)

with

q_{j} = ∫_{0}^{∞} Q^{(j)}(u)ν_{j}^{+}Q^{∗}(u) du, j = 0, …, N,

and the remainder term

ρ_{N+1}(ε, 0) = \frac{ε}{N!} \{[∫_{0}^{1} ∫_{0}^{∞} ∫_{u}^{u+εv} Q(u)R(v)Q^{N+1}(w)(u + εv − w)^{N} dw du dv]^{∗}

+ ∫_{0}^{1} ∫_{0}^{∞} ∫_{u}^{u+εv} Q(u)R(v)Q^{N+1}(w)(u + εv − w)^{N} dw du dv} .

The values for τ < 0 can be calculated from the relation ^{ε}R_{gg}(τ) = ^{ε}R_{gg}^{∗}(−τ).

Example Let us consider a system of n ordinary differential equations of first order with constant coefficients and an ε-correlated wide-sense stationary inhomogeneous term

εẋ = A^{ε}x + ^{ε}f.

The matrix A is assumed to be stable (i.e. all eigenvalues have negative real parts) and diagonalizable, i.e. A = V ΛV^{−1} with Λ = diag(λ_{1}, …, λ_{n}). Then a stationary solution

εx(t) = ∫_{−∞}^{t} e^{A(t−s)} ^{ε}f(s) ds

exists (see e.g. [1], [4], [6]). For fixed τ the asymptotic expansions can be calculated as follows.

For τ = 0 it holds

R_{gg}(0) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{q_{j} + q_{j}^{∗}\} + o(ε^{N+1})

with

q_{j} = V Λ^{j} ∫_{0}^{∞} e^{Λu}B_{j}e^{Λ^{∗}u}du V^{∗} = −V (\frac{b_{jkl}λ_{k}^{j}}{λ_{k} + \overline{λ_{l}}} )_{k,l=1,2,...,n} V^{∗},
B_{j} = (b_{jkl})_{k,l=1,2,...,n} = V^{−1}ν_{j}^{+}[V^{−1}]^{∗},

and for τ > 0 it results

R_{gg}(τ) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} q_{j}(τ) + o(ε^{N+1})

with

q_{j}(τ) = V ∫_{0}^{∞} e^{Λu}C_{j}e^{Λ^{∗}u}du [Λ^{j}]^{∗}e^{Λ^{∗}τ}V^{∗} = −V (\frac{c_{jkl}e^{\overline{λ_{l}}τ} \overline{λ_{l}}^{j}}{λ_{k} + \overline{λ_{l}}} )_{k,l=1,2,...,n}V^{∗},
C_{j} = (c_{jkl})_{k,l=1,2,...,n} = V^{−1}μ_{j}^{∗}[V^{−1}]^{∗}.

4 Expansion for scalar complex processes

In this section asymptotic expansions as ε → 0 for the special case of complex valued wide-sense stationary processes and scalar complex valued kernel functions are given. Here, the correlation functions are defined by

εR(s) = E\{^{ε}f(t) \overline{^{ε}f(t + s)}\} = R (\frac{s}{ε}) , ε > 0,

according to condition 4 of Assumption 1. Now, for the correlation moments we get

μ_{j} = ∫_{−∞}^{∞} s^{j}R(s) ds = \{ \begin{matrix}2 ∫_{0}^{∞}s^{j}ReR(s) ds & for j even, \\ 2i ∫_{0}^{∞}s^{j}ImR(s) ds & for j odd , \end{matrix}

ν_{j} = ∫_{−∞}^{∞} |s|^{j}R(s) ds = 2 ∫_{0}^{∞}s^{j}ReR(s) ds,

ν_{j}^{+} = ∫_{0}^{∞}s^{j}R(s) ds,

hence \overline{μ_{j}} = (−1)^{j}μ_{j}, \overline{ν_{j}} = ν_{j}. With analogous assumptions as in the real or vector valued case the following asymptotic expansions are valid.

• For ^{ε}r = ∫_{−∞}^{∞} Q(s)^{ε}f(s) ds it holds

E\{^{ε}r \overline{^{ε}r}\} = E\{|^{ε}r|^{2}\} = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \overline{μ_{j}}q_{j} + ρ_{N+1}(ε)

with  q_{j} = ∫_{−∞}^{∞} Q^{(j)}(t)\overline{Q(t)} dt.

• For ^{ε}r = ∫_{0}^{∞}Q(s)^{ε}f(s) ds it holds

E\{^{ε}r \overline{^{ε}r}\} = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{ν_{j}^{+}q_{j} + \overline{ν_{j}^{+}}\overline{q_{j}}\} + ρ_{N+1}(ε)
with  q_{j} = ∫_{0}^{∞} Q(t)\overline{Q^{(j)}(t)} dt.

• For ^{ε}r = ∫_{a}^{b} Q(s)^{ε}f(s) ds it holds

E\{^{ε}r \overline{^{ε}r}\} = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{ν_{j}^{+}q_{j} + \overline{ν_{j}^{+}}\overline{q_{j}}\} + ρ_{N+1}(ε)

with  q_{j} = ∫_{a}^{b} Q(t)\overline{Q^{(j)}(t)} dt − ∑_{l=0}^{j−1}(−1)^{l}Q^{(l)}(b)\overline{Q^{(j−1−l)}(b)}.

• For the correlation functions ^{ε}R_{gg}(τ) = E\{^{ε}g(t) \overline{^{ε}g(t + τ)}\} of the stationary processes

εg(t) = ∫_{−∞}^{t} Q(t − s)^{ε}f(s) ds

it holds for τ = 0

εR_{gg}(0) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{ν_{j}^{+}q_{j} + \overline{ν_{j}^{+}}\overline{q_{j}}\} + ρ_{N+1}(ε, 0)
with  q_{j} = ∫_{0}^{∞} Q^{(j)}(u)\overline{Q(u)} du,

for τ > 0 it results

εR_{gg}(τ) = ∑_{j=0}^{N} \frac{ε^{j+1}}{j!} \{\overline{μ_{j}}q_{j}(τ) + 1_{\{0<τ<ε\}}(τ)c_{j}(τ)\} + ρ_{N+1}(ε, τ)
with q_{j}(τ) = ∫_{0}^{∞} Q(u)\overline{Q^{(j)}(u + τ)} du,
c_{j}(τ) = \overline{q_{j}(τ)} ∫ 1_{\frac{τ}{ε}} (v − \frac{2τ}{ε} )^{j} R(v)dv + (−1)^{j+1}q_{j}(τ) ∫ 1_{\frac{τ}{ε}} v^{j}R(v)dv,

and the values for τ < 0 can be found from the relation ^{ε}R_{gg}(τ) = \overline{^{ε}R_{gg}(−τ)}.

5 Conclusion

The results of [7] concerning asymptotic expansions for ε → 0 of variances or correlation functions of real valued integral functionals involving ε-correlated widesense stationary random processes are extended to the case of complex vector valued random functions. These expansions can be used in the investigation of random vibrations, e.g. given by a system of ordinary differential equations with a random inhomogeneous term. In the paper also the corresponding asymptotic expansions for scalar complex valued processes are given. An extension of the results to integral functionals of scalar or vector valued random fields and certain classes of nonstationary processes will be considered in a subsequent paper.

Acknowledgment

The support by the Deutsche Forschungsgemeinschaft is greatfully appreciated.

References

[1] H. Bunke. Gewöhnliche Differentialgleichungen mit zufälligen Parametern. Akademie-Verlag, Berlin, 1972.

[2] R. Courant, and F. John. Introduction to calculus and analysis. Vol. I. Springer-Verlag, New York, 1989.

[3] I. P. Natanson. Theorie der Funktionen einer reellen Veränderlichen. Akademie-Verlag, Berlin, 1975.

[4] A. Preumont. Random vibration and spectral analysis. Kluwer, Dordrecht, 1994.

[5] J. vom Scheidt. Stochastic equations of mathematical physics. Akademie-Verlag, Berlin, 1990.

[6] J. vom Scheidt, B. Fellenberg, and U. Wöhrl. Analyse und Simulation stochastischer Schwingungssysteme. B. G. Teubner, Stuttgart, 1994.

[7] J. vom Scheidt, H.-J. Starkloff, and R. Wunderlich. Asymptotic expansions for second-order moments of integral functionals of weakly correlated random functions. Preprint 97-17, TU Chemnitz, Faculty of Mathematics, 1997, http://www.mathematik.tu-chemnitz.de/preprint.