The Extreme Terms of a Sample and Their Role in the Sum of Independent Variables

On the Extremal Theory of Continued Fractions

“…The proof shows that (1.5) remains valid if S (1) N is replaced by S (d) N for any fixed d ≥ 2 and discarding more terms improves the rate of a.e. convergence in (1.5).…”

“…variables in the domain of attraction of a stable law with parameter 0 < α < 2, the effect of the extremal terms on the partial sums is well known. For positive variables Darling [8] showed (see also Arov and Bobrov [1]) that under some additional regularity assumptions the ratio of the sum and its largest term has a non-degenerate limit distribution if 0 < α < 1 and this holds also for 1 < α < 2 provided we center the partial sum by its mean. The case α = 1 is critical and is not covered in [1], [8].…”

“…For positive variables Darling [8] showed (see also Arov and Bobrov [1]) that under some additional regularity assumptions the ratio of the sum and its largest term has a non-degenerate limit distribution if 0 < α < 1 and this holds also for 1 < α < 2 provided we center the partial sum by its mean. The case α = 1 is critical and is not covered in [1], [8]. The sequence {a k (x), k ≥ 1} in the continued fraction expansion corresponds to this case, except that the variables a k are weakly dependent.…”

“…Moreover, since U n (t, s) is constant on intervals k/n ≤ t < (k + 1)/n, by the last statement of Lemma 2.1 we may assume that nt, nt 1 and nt 2 are all integers. To prove (3.11), we introduce the notations…”

“…Clearly X (1) i and X (2) i are supported on different sets and thus X (1) i X (2) i = 0. Thus among the 9 terms above, the first, second, fourth and fifth are equal to 0.…”