A martingale approach to the PASTA property

1993 ◽  
Vol 30 (1) ◽  
pp. 252-257
Author(s):  
Michael Scheutzow
Keyword(s):  
Poisson Process ◽  
Queueing Theory ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Boundedness Condition ◽  
Martingale Approach ◽  
Large Numbers ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Image Position

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

A martingale approach to the PASTA property

1993 ◽  
Vol 30 (01) ◽  
pp. 252-257
Author(s):  
Michael Scheutzow
Keyword(s):  
Poisson Process ◽  
Queueing Theory ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Boundedness Condition ◽  
Martingale Approach ◽  
Large Numbers ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Image Position

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L 2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

ASYMPTOTIC BEHAVIORS FOR CORRELATED BERNOULLI MODEL

2019 ◽  
Vol 34 (4) ◽  
pp. 570-582
Author(s):  
Yu Miao ◽  
Huanhuan Ma ◽  
Qinglong Yang
Keyword(s):  
Law Of Large Numbers ◽  
Moderate Deviation ◽  
Bernoulli Model ◽  
Strong Law ◽  
Asymptotic Behaviors ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Large Numbers ◽  
Bernoulli Variables ◽  
Image Position

AbstractWe consider a class of correlated Bernoulli variables, which have the following form: for some 0 < p < 1, $$\begin{align}{P(X_{j+1}=1 \vert {\cal F}_{j})= (1-\theta_j)p+\theta_jS_j/j,}\end{align}$$where 0 ≤ θj ≤ 1, $S_n=\sum _{j=1}^nX_j$ and ${\cal F}_n=\sigma \{X_1,\ldots , X_n\}$. The aim of this paper is to establish the strong law of large numbers which extend some known results, and prove the moderate deviation principle for the correlated Bernoulli model.

Asymptotic analysis for affine point processes with large initial intensity

2018 ◽  
Vol 17 (01) ◽  
pp. 117-143
Author(s):  
Nian Yao ◽  
Mingqing Xiao
Keyword(s):  
Asymptotic Behavior ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
General Structure ◽  
Large Numbers ◽  
Special Cases ◽  
Classical Models ◽  
Initial Intensity

In this paper, we consider a generalized stochastic model associated with affine point processes based on several classical models. In particular, we study the asymptotic behavior of the process when the initial intensity is large, i.e. the intensity of arriving events observed initially is considerably larger, which appears in many real applications. For our generalized model, we establish (i) the large deviation principle; (ii) the corresponding functional law of large numbers; (iii) the corresponding central limit theorem, that reflect the fundamentals of the process asymptotic behavior. Our obtained results include existing results as special cases with a more general structure.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

2015 ◽  
Vol 52 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Raúl Fierro ◽  
Víctor Leiva ◽  
Jesper Møller
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Hawkes Process ◽  
Homogeneous Poisson Process ◽  
Large Numbers ◽  
Main Shocks

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (3) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

Exact convergence rate of an Erdös-Rényi strong law for moving quantiles

1986 ◽  
Vol 23 (02) ◽  
pp. 355-369
Author(s):  
Paul Deheuvels ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Order Statistic ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Monotonicity Condition ◽  
Strict Monotonicity ◽  
Strong Law ◽  
Large Numbers ◽  
Image Position

Consider a sequence U 1, U 2 , · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un +1, · ··, Un +K , and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α &lt; u &lt; 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X 1, X 2, · ·· with d.f. F satisfying a strict monotonicity condition.

Exact convergence rate of an Erdös-Rényi strong law for moving quantiles

1986 ◽  
Vol 23 (2) ◽  
pp. 355-369 ◽  
Author(s):  
Paul Deheuvels ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Order Statistic ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Monotonicity Condition ◽  
Strict Monotonicity ◽  
Strong Law ◽  
Large Numbers ◽  
Image Position

Consider a sequence U1, U2, · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un+1, · ··, Un+K, and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α < u < 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X1, X2, · ·· with d.f. F satisfying a strict monotonicity condition.

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