On Limit Theorems for the Sum of a Random Number of Random Summands with Values in LCA-Groups*

2013 ◽  
Vol 196 (1) ◽  
pp. 54-56
Author(s):  
E. M. Kudlaev
Keyword(s):  
Limit Theorems ◽  
Random Number ◽  
Lca Groups

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

2000 ◽  
Vol 32 (01) ◽  
pp. 159-176 ◽  
Author(s):  
Markus Bachmann
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Number ◽  
Time Lag ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Branching Random Walk ◽  
Minimal Position ◽  
Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

Limit theorems for thinning of renewal point processes

1972 ◽  
Vol 9 (4) ◽  
pp. 847-851 ◽  
Author(s):  
Råde L.
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Number

Limit theorems for the thinning of renewal point processes according to two different schemes are studied. In the first scheme when a point is retained a random number of succeeding points are deleted. According to the second scheme a random number of points are deleted by an inhibitory Poisson process.

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.

Asymptotic approximations for coupon collectors

2009 ◽  
Vol 46 (1) ◽  
pp. 61-96
Author(s):  
Anna Pósfai ◽  
Sándor Csörgő
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Random Number ◽  
Limiting Distribution ◽  
Asymptotic Approximations ◽  
The One ◽  
First Time

A collector samples with replacement a set of n ≧ 2 distinct coupons until he has n − m , 0 ≦ m < n , distinct coupons for the first time. We refine the limit theorems concerning the standardized random number of necessary draws if n → ∞ and m is fixed: we give a one-term asymptotic expansion of the distribution function in question, providing a better approximation of it, than the one given by the limiting distribution function, and proving in particular that the rate of convergence in these limiting theorems is of order (log n )/ n .

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 for the minimal position in a branching random walk with independent logconcave displacements

2000 ◽  
Vol 32 (1) ◽  
pp. 159-176 ◽  
Author(s):  
Markus Bachmann
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Number ◽  
Time Lag ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Branching Random Walk ◽  
Minimal Position ◽  
Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

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