scholarly journals Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Cristian Giardinà ◽  
Claudio Giberti ◽  
Elena Magnanini
Keyword(s):  
Random Graph ◽  
Generating Function ◽  
Cumulant Generating Function

The Cumulant Generating Function Estimation Method

1997 ◽  
Vol 13 (2) ◽  
pp. 170-184 ◽  
Author(s):  
John L. Knight ◽  
Stephen E. Satchell
Keyword(s):  
Characteristic Function ◽  
Density Function ◽  
Generating Function ◽  
Estimation Method ◽  
Empirical Characteristic Function ◽  
Function Estimation ◽  
Asymptotic Equivalence ◽  
Cumulant Generating Function ◽  
Worked Example ◽  
Step Procedure

This paper deals with the use of the empirical cumulant generating function to consistently estimate the parameters of a distribution from data that are independent and identically distributed (i.i.d.). The technique is particularly suited to situations where the density function is unknown or unbounded in parameter space. We prove asymptotic equivalence of our technique to that of the empirical characteristic function and outline a six-step procedure for its implementation. Extensions of the approach to non-i.i.d. situations are considered along with a discussion of suitable applications and a worked example.

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

scholarly journals New perspectives on the Erlang-A queue

2019 ◽  
Vol 51 (01) ◽  
pp. 268-299 ◽  
Author(s):  
Andrew Daw ◽  
Jamol Pender
Keyword(s):  
Generating Function ◽  
Large Scale ◽  
Moment Generating Function ◽  
Urban Mobility ◽  
Cumulant Generating Function ◽  
True Model ◽  
The Difference ◽  
The Moment ◽  
Stochastic Representations ◽  
Transient And Steady State

AbstractThe nonstationary Erlang-A queue is a fundamental queueing model that is used to describe the dynamic behavior of large-scale multiserver service systems that may experience customer abandonments, such as call centers, hospitals, and urban mobility systems. In this paper we develop novel approximations to all of its transient and steady state moments, the moment generating function, and the cumulant generating function. We also provide precise bounds for the difference of our approximations and the true model. More importantly, we show that our approximations have explicit stochastic representations as shifted Poisson random variables. Moreover, we are also able to show that our approximations and bounds also hold for nonstationary Erlang-B and Erlang-C queueing models under certain stability conditions.

Asymptotics of Implied Volatility far from Maturity

2009 ◽  
Vol 46 (03) ◽  
pp. 629-650 ◽  
Author(s):  
Michael R. Tehranchi
Keyword(s):  
Generating Function ◽  
Stock Price ◽  
Implied Volatility ◽  
Call Option ◽  
Precise Control ◽  
Cumulant Generating Function ◽  
European Call Option ◽  
Asymptotic Formulae ◽  
Error Terms

This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.

Limiting distributions for the general branching process with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 809-813 ◽  
Author(s):  
S. D. Durham
Keyword(s):  
Generating Function ◽  
Age Structure ◽  
Branching Process ◽  
Critical Case ◽  
Functional Form ◽  
Asymptotic Distributions ◽  
Moment Conditions ◽  
Cumulant Generating Function ◽  
Age Dependent ◽  
Branching Process With Immigration

For the general age-dependent branching process subject to an arbitrary pattern of immigration asymptotic distributions are studied in the supercritical and critical cases. In the supercritical case a new functional form is given for the limiting cumulant-generating function and in the critical case moment conditions are given under which a randomized gamma limit law prevails. The age-structure is also considered.

Limiting distributions for the general branching process with immigration

1974 ◽  
Vol 11 (04) ◽  
pp. 809-813
Author(s):  
S. D. Durham
Keyword(s):  
Generating Function ◽  
Age Structure ◽  
Branching Process ◽  
Critical Case ◽  
Functional Form ◽  
Asymptotic Distributions ◽  
Moment Conditions ◽  
Cumulant Generating Function ◽  
Age Dependent ◽  
Branching Process With Immigration

For the general age-dependent branching process subject to an arbitrary pattern of immigration asymptotic distributions are studied in the supercritical and critical cases. In the supercritical case a new functional form is given for the limiting cumulant-generating function and in the critical case moment conditions are given under which a randomized gamma limit law prevails. The age-structure is also considered.

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