On limit theorems for a random number of random variables

1983 ◽  
pp. 167-176 ◽  
Author(s):  
B. V. Gnedenko
Keyword(s):  
Limit Theorems ◽  
Random Number ◽  
Random Variables

Limit theorems and estimation theory for branching processes with an increasing random number of ancestors

1997 ◽  
Vol 34 (2) ◽  
pp. 309-327 ◽  
Author(s):  
J. P. Dion ◽  
N. M. Yanev
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Number ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Transfer Theorem ◽  
Asymptotically Normal ◽  
The Mean ◽  
Watson Process

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

scholarly journals Lifetime Distributions and their Approximation in Reliability of Serial/Parallel Networks

2020 ◽  
Vol 28 (2) ◽  
pp. 161-172
Author(s):  
Alexei Leahu ◽  
Veronica Andrievschi-Bagrin
Keyword(s):  
Power Series ◽  
Limit Theorems ◽  
Random Number ◽  
Random Variables ◽  
Lifetime Distributions ◽  
Parallel Networks ◽  
Independent Identically Distributed

AbstractIn this paper we present limit theorems for lifetime distributions connected with network’s reliability as distributions of random variables(r.v.) min(Y1, Y2,..., YM) and max(Y1, Y2,..., YM ), where Y1, Y2,..., are independent, identically distributed random variables (i.i.d.r.v.), M being Power Series Distributed (PSD) r.v. independent of them and, at the same time, Yk, k = 1, 2, ..., being a sum of non-negative, i.i.d.r.v. in a Pascal distributed random number.

Limit theorems and estimation theory for branching processes with an increasing random number of ancestors

1997 ◽  
Vol 34 (02) ◽  
pp. 309-327 ◽  
Author(s):  
J. P. Dion ◽  
N. M. Yanev
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Number ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Transfer Theorem ◽  
Asymptotically Normal ◽  
The Mean ◽  
Watson Process

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

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).

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