Local limit theorems for non-critical Galton–Watson processes by or without immigration

1982 ◽  
Vol 19 (02) ◽  
pp. 262-271
Author(s):  
R. Höpfner
Keyword(s):  
Limit Theorems ◽  
Critical Case ◽  
Transition Probabilities ◽  
Local Limit ◽  
Limiting Distributions ◽  
Initial State ◽  
Final State ◽  
Initial States ◽  
Local Limit Theorems

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.

Local limit theorems for non-critical Galton–Watson processes by or without immigration

1982 ◽  
Vol 19 (2) ◽  
pp. 262-271 ◽  
Author(s):  
R. Höpfner
Keyword(s):  
Limit Theorems ◽  
Critical Case ◽  
Transition Probabilities ◽  
Local Limit ◽  
Limiting Distributions ◽  
Initial State ◽  
Final State ◽  
Initial States ◽  
Local Limit Theorems

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilities p0,1 = 1, pk,k–1 = gk, pk,k+1 = 1– gk (0 < gk < 1, k = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilitiesp0,1= 1,pk,k–1=gk,pk,k+1= 1–gk(0 &lt;gk&lt; 1,k= 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

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.

Large deviations of generalized renewal process

2020 ◽  
Vol 30 (4) ◽  
pp. 215-241
Author(s):  
Gavriil A. Bakay ◽  
Aleksandr V. Shklyaev
Keyword(s):  
Large Deviations ◽  
Renewal Process ◽  
Limit Theorems ◽  
Local Limit ◽  
Asymptotic Formulas ◽  
Additive Functionals ◽  
Wide Range ◽  
Finite Markov Chains ◽  
Independent Identically Distributed ◽  
Local Limit Theorems

AbstractLet (ξ(i), η(i)) ∈ ℝd+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, NT = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process ZT = $\begin{array}{} \sum_{i=1}^{N_T} \end{array}$ξ(i). Put IΔT(x) = {y ∈ ℝd : xj ≤ yj < xj + ΔT, j = 1, …, d}. We find asymptotic formulas for the probabilities P(ZT ∈ IΔT(x)) as ΔT → 0 and P(ZT = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

Local limit theorems for generalized scheme of allocation of particles into ordered cells

2021 ◽  
Vol 31 (4) ◽  
pp. 293-307
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Weak Convergence ◽  
Negative Binomial Distribution ◽  
Limit Theorems ◽  
Negative Binomial ◽  
Sufficient Conditions ◽  
Local Limit ◽  
Number Of Components ◽  
Normal Law ◽  
Local Limit Theorems ◽  
Generalized Scheme

Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.

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