The limit distribution of the length of the longest head-run

1979 ◽  
Vol 10 (4) ◽  
pp. 301-310 ◽  
Author(s):  
Antónia Földes
Keyword(s):  
Limit Distribution ◽  
Head Run

The A.S. Limit Distribution of the Longest Head Run

1993 ◽  
Vol 45 (6) ◽  
pp. 1245-1262 ◽  
Author(s):  
Tamás F. Móri
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Distribution ◽  
Remainder Term ◽  
Fair Coin ◽  
Head Run ◽  
Stochastically Bounded ◽  
General Setup

AbstractIt is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2n with a stochastically bounded remainder term. Though — log2n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.

On the first zero of an empirical characteristic function

1991 ◽  
Vol 28 (3) ◽  
pp. 593-601 ◽  
Author(s):  
H. U. Bräker ◽  
J. Hüsler
Keyword(s):  
Characteristic Function ◽  
Limit Distribution ◽  
Random Variable ◽  
Empirical Characteristic Function ◽  
The Real ◽  
Exponential Decrease ◽  
Characteristic Process

We deal with the distribution of the first zero Rn of the real part of the empirical characteristic process related to a random variable X. Depending on the behaviour of the theoretical real part of the underlying characteristic function, cases with a slow exponential decrease to zero are considered. We derive the limit distribution of Rn in this case, which clarifies some recent results on Rn in relation to the behaviour of the characteristic function.

scholarly journals Non-local Solvable Birth–Death Processes

2021 ◽  
Author(s):  
Giacomo Ascione ◽  
Nikolai Leonenko ◽  
Enrica Pirozzi
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Limit Distribution ◽  
Spectral Decomposition ◽  
Strong Solutions ◽  
Stochastic Representation ◽  
Discrete Approximations ◽  
Local Difference ◽  
Non Local ◽  
Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

Threshold limit theorems for some epidemic processes

1980 ◽  
Vol 12 (02) ◽  
pp. 319-349 ◽  
Author(s):  
Bengt Von Bahr ◽  
Anders Martin-Löf
Keyword(s):  
Limit Distribution ◽  
Limit Theorems ◽  
Polymerization Process ◽  
Connected Components ◽  
Critical Threshold ◽  
Infected Person ◽  
Initial Number ◽  
Total Size ◽  
Frost Model ◽  
Healthy Persons

The Reed–Frost model for the spread of an infection is considered and limit theorems for the total size, T, of the epidemic are proved in the limit when n, the initial number of healthy persons, is large and the probability of an encounter between a healthy and an infected person per time unit, p, is λ/n. It is shown that there is a critical threshold λ = 1 in the following sense, when the initial number of infected persons, m, is finite: If λ ≦ 1, T remains finite and has a limit distribution which can be described. If λ > 1 this is still true with a probability σ m < 1, and with probability 1 – σ m T is close to n(1 – σ) and has an approximately Gaussian distribution around this value. When m → ∞ also, only the Gaussian part of the limit distribution is obtained. A randomized version of the Reed–Frost model is also considered, and this allows the same result to be proved for the Kermack–McKendrick model. It is also shown that the limit theorem can be used to study the number of connected components in a random graph, which can be considered as a crude description of a polymerization process. In this case polymerization takes place when λ > 1 and not when λ ≦ 1.

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