Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

A Pascal-like Triangle Related to the Tribonacci Numbers

A tile composed of two pieces, which we refer to as a fence tile, is introduced to give the Tribonacci numbers a new combinatorial interpretation. The interpretation is used to construct a Pascal-like triangle and various identities concerning the triangle are proven. An intuitive proof of a general identity for the Tribonacci numbers in terms of sums of products of the binomial coefficients is...

Central and local limit theorems for RNA structures.

A k-noncrossing RNA pseudoknot structure is a graph over {1,...,n} without 1-arcs, i.e. arcs of the form (i,i+1) and in which there exists no k-set of mutually intersecting arcs. In particular, RNA secondary structures are 2-noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the number of 3-noncrossing RNA structures over n nucleotides...

From Fibonacci numbers to central limit type theorems

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}n=1. Lekkerkerker proved that the average number of summands for integers in [Fn, Fn+1) is n/(φ 2+1), with φ the golden mean. This has been generalized to the following: given nonnegative integers c1, c2, . . . , cL with c1, cL > 0 and recursive sequence {Hn}n=1 ...

Central and Local Limit Theorems in Markov Dependent Random Variables

We consider an irreducible and aperiodic Markov chain {kn}n=0 over the finite state space E = {1, . . . , p} with positive regular transition matrix P = {pij} and additive component {Un} such that {Sn} = {(kn, Un)} is also a Markov chain over the state space E1 = E × R. We prove a central and a local limit theorem for this chain when the probability density functions of {Sn}, conditional on the...

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