An Estimate of the Remainder Term in Limit Theorems of Renewal Theory

1974 ◽  
Vol 18 (4) ◽  
pp. 662-677 ◽  
Author(s):  
B. A. Rogozin
Keyword(s):  
Remainder Term ◽  
Limit Theorems ◽  
Renewal Theory

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (04) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

A Central Limit Theorem for Cumulative Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (4) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

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