scholarly journals Limit Theorems for Probabilities of Large Deviations of a Critical Galton–Watson Process Having Power Tails

2008 ◽  
Vol 52 (4) ◽  
pp. 674-688 ◽  
Author(s):  
V. I. Wachtel
Keyword(s):  
Large Deviations ◽  
Limit Theorems ◽  
Probabilities Of Large Deviations ◽  
Watson Process

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.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

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