scholarly journals The distribution of m-ary search trees generated by van der Corput sequences

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Wolfgang Steiner
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Function Approach ◽  
Search Trees ◽  
Asymptotically Normal

We study the structure of $m$-ary search trees generated by the van der Corput sequences. The height of the tree is calculated and a generating function approach shows that the distribution of the depths of the nodes is asymptotically normal. Additionally a local limit theorem is derived.

Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (04) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

scholarly journals Locally Restricted Compositions I. Restricted Adjacent Differences

10.37236/1954 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Edward A. Bender ◽  
E. Rodney Canfield
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Sufficient Conditions ◽  
Local Limit ◽  
The Other ◽  
Simple Condition ◽  
Suitable Constant ◽  
Distinct Part ◽  
Positive Elements

We study compositions of the integer $n$ in which the first part, successive differences, and the last part are constrained to lie in prescribed sets ${\cal L,D,R}$, respectively. A simple condition on ${\cal D}$ guarantees that the generating function $f(x,{\cal L,D,R})$ has only a simple pole on its circle of convergence and this at $r({\cal D})$, a function independent of ${\cal L}$ and ${\cal R}$. Thus the number of compositions is asymptotic to $Ar({\cal D})^{-n}$ for a suitable constant $A=A({\cal L,D,R})$. We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of $n$, such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If ${\cal D}$ has only finitely many positive elements ${\cal D}^+$, or finitely many negative elements ${\cal D}^-$, then the largest part and number of distinct part sizes are almost surely $\Theta((\log n)^{1/2})$. On the other hand, when both ${\cal D}^+$ and ${\cal D}^-$ have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely $\Theta(\log n)$, and we give sufficient conditions for the largest part to be almost surely asymptotic to $\log_{1/r({\cal D})}n$.

scholarly journals Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (4) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

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