  We prove that the Birkhoff sum S(n)/n = (1/n) sum_(k=1)^(n-1) g(k A) with
g(x) = cot(Pi x) and golden ratio A converges in the sense that the sequence of
functions s(x) = S([ x q(2n)])/q(2n) with Fibonacci numbers q(n) converges to a
self similar limiting function s(x) on [0,1]. The function s(x) can be computed
analytically. This allows to determine values like S(10^100)/10^100 accurately
without that it ever would be possible to add up so many terms for this random
walk. The random variables added up are Cauchy distributed random variables
with almost periodic correlation. While for any continuous function g, the
Birkhoff limiting function is s(x)=M x by Birkhoff's ergodic theorem, we get so
examples of random variables X(n), where the limiting function of S([x n])/n
converges to a nontrivial selfsimilar function s(x) along subsequences for one
initial point. Hardy and Littlewood have studied the Birkhoff sum for g'(x)=
-Pi csc^2(Pi x) and shown that the corresponding Birkhoff sum S'(n)/n^2 stays
bounded. Sinai and Ulcigrai have found a limiting distribution for g(x) if both
the rotation number A and the initial point t are integrated over. We fix the
golden ratio A, start with fixed t=0 and show that the rescaled random walk
converges along subsequences. The analysis sheds light on the pictures seen in
previous papers with Lesieutre and Tangerman on the antiderivative G(x).
