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

86 A limit theorem for the Hurwitz zeta-function...
where X n is an H(D)-valued random element having the distribution P 1 n.
Let {K l } be a sequence of compact subsets of D such that
∞⋃
K l = D,
l=1
K l ⊂ K l+1 , and if K is a compact of D, then K ⊆ K l for some l. Define
ρ(f, g) =
∞∑
l=1
2 −l ρ l (f, g)
, f, g ∈ H(D),
1 + ρ l (f, g)
where ρ l (f, g) = sup
s∈K l
|f(s) − g(s)|. Then ρ is a metric on H(D) which induces its
topology.
Since the series for ζ 2,n (s, α) is absolutely convergent in D, we have
sup
lim
n1 T →∞
1
T
∫ T
0
sup |ζ 2,n (s + iτ, α)|dτ R l < ∞. (16)
s∈K l
Let ε > 0 and M l = R l2 l
ε
. Then by the Chebyshev inequality
P ( )
sup |X T,n (s, α)| > M 1
l
s∈K l
T M l
∫T
0
sup |ζ 2,n (s + iτ, α)|dτ,
s∈K l
and, consequently,
Thus we find that
and, for all n 1,
lim P( )
sup |X T,n (s, α)| > M ε l
T →∞ s∈K l
2 l .
P ( )
sup |X n (s, α)| > M ε l
s∈K l
2 l ,
P ( X n (s, α) ∈ H ε
)
1 − ε
or, since P 1 n is the distribution of X n ,
P 1 n(H ε ) 1 − ε,
where H ε = {h ∈ H(D) : sup |h(s)| M l , l 1}. The set H ε is compact.
s∈K l
Therefore the family of the measures {Pn} 1 is tight, and by the Prokhorov theorem
it is relatively compact.