The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integral orders

1981 ◽  
Vol 102 (1) ◽  
pp. 79-105 ◽  
Author(s):  
Albrecht Böttcher ◽  
Bernd Silbermann
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Toeplitz Determinants

scholarly journals A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large

10.37236/1517 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Asymptotic Behavior ◽  
Saddle Point ◽  
Generating Functions ◽  
Single Point ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Point Method ◽  
Saddle Point Method ◽  
Numerical Verification ◽  
Height Distribution

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to $b$ items (the so called $b$-tries). We assume that such a tree is built from $n$ random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when $b$ and $n$ are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed $b$. We prove that for most $n$, the limiting distribution is concentrated at the single point $k_1=\lfloor \log_2 (n/b)\rfloor +1$ as $n,b\to \infty$. We observe that this is quite different than the height distribution for fixed $b$, in which case the limiting distribution is of an extreme value type concentrated around $(1+1/b)\log_2 n$. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.

scholarly journals A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL M. KANE ◽  
ROBERT C. RHOADES
Keyword(s):  
Asymptotic Behavior ◽  
Modular Form ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Bootstrap Percolation ◽  
Asymptotic Result ◽  
Integer Partitions ◽  
Mock Modular Form ◽  
Precise Asymptotic

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

UNIT ROOT SEASONAL AUTOREGRESSIVE MODELS WITH A POLYNOMIAL TREND OF HIGHER DEGREE

2001 ◽  
Vol 17 (2) ◽  
pp. 357-385 ◽  
Author(s):  
Seiji Nabeya
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Unit Root ◽  
Autoregressive Models ◽  
Sample Period ◽  
Limiting Distributions ◽  
Polynomial Degree ◽  
Polynomial Trend ◽  
Autoregressive Parameter ◽  
Simulation Results

Seasonal autoregressive models with a polynomial trend of higher degee are treated. In the unit root case, the limiting distribution of the normalized least squares estimator for the autoregressive parameter and that of the corresponding t-statistic are discussed as the length of the sample period tends to infinity. In the case where the polynomial trend has the second or third degree, the joint moment generating functions associated with these limiting distributions are derived, and some simulation results are reported. The asymptotic behavior of these limiting distributions is discussed when the polynomial degree or the number of seasons tends to infinity.

ASYMPTOTICS OF A TAU-FUNCTION AND TOEPLITZ DETERMINANTS WITH SINGULAR GENERATING FUNCTIONS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 83-107 ◽  
Author(s):  
ESTELLE L. BASOR ◽  
CRAIG A. TRACY
Keyword(s):  
Generating Functions ◽  
Short Distance ◽  
Quantum Fields ◽  
Toeplitz Determinants ◽  
Tau Function

We compute the short distance asymptotics of a tau-fucntion appearing in the work of Sato, Miwa, and Jimbo on holonomic quantum fields, We show that these asymptotics are determined by the Widom operator. This same operator is fundamental in the asymptotics of Toeplitz determinants with singular generating fucntions.

Sign in / Sign up

Export Citation Format

Share Document