FINITE HORIZON RUIN PROBABILITIES FOR RANDOM WALKS WITH HEAVY TAILED INCREMENTS

2013 ◽  
Vol 27 (2) ◽  
pp. 237-246
Author(s):  
Yingdong Lu
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Optimal Stopping ◽  
Fourier Transformation ◽  
Main Idea ◽  
Finite Horizon ◽  
Ruin Probabilities ◽  
Heavy Tailed

We study the asymptotic behavior of finite horizon ruin probabilities for random walks with heavy tailed increment via corrected diffusion approximation. We follow the main idea in [4] of inverting Fourier transformation, and the Fourier transformation is calculated through optimal stopping and a central limit theorem for renewal process.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

From the central limit theorem to heavy-tailed distributions

2003 ◽  
Vol 40 (3) ◽  
pp. 803-806 ◽  
Author(s):  
Jinwen Chen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Power Law ◽  
Power Law Distribution ◽  
The Central Limit Theorem ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Random Indices ◽  
Power Law Distributions

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

scholarly journals Asymptotic Behavior of Ruin Probabilities in an Insurance Risk Model with Quasi-Asymptotically Independent or Bivariate Regularly Varying-Tailed Main Claim and By-Claim

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Yang Yang ◽  
Xinzhi Wang ◽  
Xiaonan Su ◽  
Aili Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Risk Model ◽  
Dependence Structure ◽  
Ruin Probabilities ◽  
Insurance Risk ◽  
Main Claim ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Insurance Risk Model

This paper considers a by-claim risk model under the asymptotical independence or asymptotical dependence structure between each main claim and its by-claim. In the presence of heavy-tailed main claims and by-claims, we derive some asymptotic behavior for ruin probabilities.

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

2004 ◽  
Vol 41 (01) ◽  
pp. 83-92 ◽  
Author(s):  
Jean Bérard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Space Time ◽  
Nearest Neighbour ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Almost All

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.

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