A note on cumulative shock models

1988 ◽  
Vol 25 (01) ◽  
pp. 220-223 ◽  
Author(s):  
Kevin K. Anderson
Keyword(s):  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Shock Model ◽  
Time Intervals ◽  
Stable Law ◽  
Shock Models ◽  
Weak Limit Theorems ◽  
First Time ◽  
Intershock Interval

A shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered. Weak limit theorems are established for the cumulative magnitude of the shocks and the first time the cumulative magnitude exceeds z without any assumption on the dependence between the intershock interval and shock magnitude.

A note on cumulative shock models

1988 ◽  
Vol 25 (1) ◽  
pp. 220-223 ◽  
Author(s):  
Kevin K. Anderson
Keyword(s):  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Shock Model ◽  
Time Intervals ◽  
Stable Law ◽  
Shock Models ◽  
Weak Limit Theorems ◽  
First Time ◽  
Intershock Interval

A shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered. Weak limit theorems are established for the cumulative magnitude of the shocks and the first time the cumulative magnitude exceeds z without any assumption on the dependence between the intershock interval and shock magnitude.

Limit theorems for general shock models with infinite mean intershock times

1987 ◽  
Vol 24 (02) ◽  
pp. 449-456 ◽  
Author(s):  
Kevin K. Anderson
Keyword(s):  
Limit Theorems ◽  
Maximum Magnitude ◽  
Shock Model ◽  
Time Intervals ◽  
Shock Models ◽  
First Time

A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.

Limit theorems for general shock models with infinite mean intershock times

1987 ◽  
Vol 24 (2) ◽  
pp. 449-456 ◽  
Author(s):  
Kevin K. Anderson
Keyword(s):  
Limit Theorems ◽  
Maximum Magnitude ◽  
Shock Model ◽  
Time Intervals ◽  
Shock Models ◽  
First Time

A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.

scholarly journals Limit theorems for additive functionals of continuous time random walks

2020 ◽  
pp. 1-22
Author(s):  
Yuri Kondratiev ◽  
Yuliya Mishura ◽  
Georgiy Shevchenko
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Additive Functional ◽  
Stable Law ◽  
Additive Functionals ◽  
Lévy Motion ◽  
Continuous Time Random Walks ◽  
Poisson Shot Noise

Abstract For a continuous-time random walk X = {X t , t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

Generalized Lorenz curves and convexifications of stochastic processes

2003 ◽  
Vol 40 (04) ◽  
pp. 906-925 ◽  
Author(s):  
Youri Davydov ◽  
Ričardas Zitikis
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Asymptotic Theory ◽  
Fixed Time ◽  
Weak Limit ◽  
Time Intervals ◽  
Lorenz Curves ◽  
Stationary Increments ◽  
Weak Limit Theorems ◽  
Vervaat Process

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

Generalized Lorenz curves and convexifications of stochastic processes

2003 ◽  
Vol 40 (4) ◽  
pp. 906-925 ◽  
Author(s):  
Youri Davydov ◽  
Ričardas Zitikis
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Asymptotic Theory ◽  
Fixed Time ◽  
Weak Limit ◽  
Time Intervals ◽  
Lorenz Curves ◽  
Stationary Increments ◽  
Weak Limit Theorems ◽  
Vervaat Process

We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

scholarly journals Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

2020 ◽  
Vol 22 (4) ◽  
pp. 415-421
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien ◽  
Nguyen Tan Nhut
Keyword(s):  
Limit Theorems ◽  
Negative Binomial ◽  
Convergence Rates ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Weak Limit ◽  
Probability Metric ◽  
Weak Limit Theorems ◽  
Distribution Theorem ◽  
Binomial Sums

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric. Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Stable Process ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Limit ◽  
Stable Law ◽  
Partial Sum Process ◽  
Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

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