Weakening the independence assumption on polar components: limit theorems for generalized elliptical distributions

2016 ◽  
Vol 53 (1) ◽  
pp. 130-145 ◽  
Author(s):  
Miriam Isabel Seifert
Keyword(s):  
Limit Theorems ◽  
Radial Component ◽  
Dependence Structure ◽  
Independence Assumption ◽  
Geometrical Meaning ◽  
Polar Components ◽  
Angular Component ◽  
Conditional Limit Theorems ◽  
Extreme Behavior ◽  
Conditional Limit

Abstract By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (1) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (01) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (2) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

The stable pedigrees of critical branching populations

1984 ◽  
Vol 21 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Olle Nerman
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Branching Laws ◽  
Conditional Limit Theorems ◽  
First Results ◽  
Conditional Limit

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

The stable pedigrees of critical branching populations

1984 ◽  
Vol 21 (03) ◽  
pp. 447-463 ◽  
Author(s):  
Olle Nerman
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Branching Laws ◽  
Conditional Limit Theorems ◽  
First Results ◽  
Conditional Limit

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

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