a limit theorem for the hurwitz zeta-function in the space of analytic ...

R. Macaitienė 79
Proof. Let the function h 1 : Ω n → Ω n be defined by the formula
h 1 (x 0 , . . . , x n ) = (x 0 e −iθ 0
, . . . , x n e −iθ n
),
where θ m = arg g(m), m = 0, . . . , n. By Lemma 1 the probability measures P n,pn
and ˜P n,pn converges weakly to the measures m nH h −1 and m nH˜h−1 , respectively,
where the function ˜h is similar to h. It is easy to see that
˜h(x0 , . . . , x n ) =
Consequently,
n∑
m=0
g(m)
(m + α) s x m
=
n∑
m=0
e iθm
(m + α)x m
= h(h 1 (x 0 , . . . , x n )).
m nH˜h−1 = m nH (h(h 1 )) −1 = m nH (h −1
1 )h−1 . (3)
Since the Haar measure m nH is invariant with respect to translation by points in Ω,
we have that m nH h −1
1 = m nH . It follows from this and from (3) that m nH˜h−1 =
m nH h −1
1 . This proves the lemma.
3. Aplication of the ergodic theory
Denote by D a compact Abelian topological group. Let m D be the Haar measure
on (D, B(D)). A one-parameter family {ϕ τ : τ ∈ R} of transformations on D is
called a one-parameter group of transformations if for τ, τ 1 , τ 2 ∈ R, g ∈ D, we have
ϕ τ1 +τ 2
(g) = ϕ τ1 (ϕ τ2 (g)) and ϕ −τ (g) = ϕ −1
τ (g).
Let a τ = {(m + α) −iτ : 0 < α < 1}, τ ∈ R, m ∈ N 0 , N 0 = N ∪ {0}. Then
{a τ : τ ∈ R} is a one-parameter group. We define the one-parameter family
{ϕ τ : τ ∈ R} of transformations on Ω taking ϕ τ (ω) = a τ ω, ω ∈ Ω. Then we have
that {ϕ τ : τ ∈ R} is a one-parameter group of measurable transformations on Ω.
Lemma 3. The one-parameter group {ϕ τ : τ ∈ R} is ergodic.
Proof is similar to that from [3].
Denote by EX the mean of the random element X.
Lemma 4. Let a process X(τ, ω) be ergodic, E|X(τ, ω)| < ∞ and let sample paths
be integrable almost surely in the Riemann sense over every finite interval. Then
almost surely.
Proof can be found in [2].
∫
1
T
lim X(τ, ω)dτ = EX(0, ω)
T →∞ T
0