Geometric renewal convergence rates from hazard rates

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

Download Full-text

RENEWAL CONVERGENCE RATES FOR DHR AND NWU LIFETIMES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964802161055 ◽

2002 ◽

Vol 16 (1) ◽

pp. 67-84 ◽

Cited By ~ 11

Author(s):

Kenneth S. Berenhaut ◽

Robert Lund

Keyword(s):

Power Series ◽

Convergence Rate ◽

Markov Chains ◽

Generating Functions ◽

Hazard Rate ◽

Convergence Rates ◽

Renewal Sequence ◽

Geometric Convergence ◽

Power Series Methods

This article studies the geometric convergence rate of a discrete renewal sequence with decreasing hazard rate or, more generally, new worse than used lifetimes. Several variants of these structural orderings are considered. The results are derived from power series methods; roots of generating functions are the prominent issue. Optimality of the rates are considered. Examples demonstrating the utility of the results, as well as applications to Markov chains, are presented.

Download Full-text

Shapes of stationary autocovariances

Journal of Applied Probability ◽

10.1017/s0021900200002539 ◽

2006 ◽

Vol 43 (04) ◽

pp. 1186-1193

Author(s):

Robert Lund ◽

Ying Zhao ◽

Peter C. Kiessler

Keyword(s):

Time Series ◽

Hazard Rate ◽

Convergence Rates ◽

Stationary Time Series ◽

Time Series Models ◽

Mean Squared Errors ◽

Geometric Convergence ◽

One Step ◽

New Better Than Used ◽

Better Than

This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.

Download Full-text

Shapes of stationary autocovariances

Journal of Applied Probability ◽

10.1239/jap/1165505218 ◽

2006 ◽

Vol 43 (4) ◽

pp. 1186-1193

Author(s):

Robert Lund ◽

Ying Zhao ◽

Peter C. Kiessler

Keyword(s):

Time Series ◽

Hazard Rate ◽

Convergence Rates ◽

Stationary Time Series ◽

Time Series Models ◽

Mean Squared Errors ◽

Geometric Convergence ◽

One Step ◽

New Better Than Used ◽

Better Than

This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.

Download Full-text

A monotonicity in reversible Markov chains

Journal of Applied Probability ◽

10.1239/jap/1152413736 ◽

2006 ◽

Vol 43 (2) ◽

pp. 486-499 ◽

Cited By ~ 2

Author(s):

Robert Lund ◽

Ying Zhao ◽

Peter C. Kiessler

Keyword(s):

Convergence Rate ◽

Markov Chains ◽

State Space ◽

Hazard Rate ◽

Probability Generating Function ◽

Countable State Space ◽

Return Times ◽

Geometric Convergence ◽

Reversible Markov Chains ◽

Countable State

In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times to every state in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.

Download Full-text

On unbounded hazard rates for smoothed perturbation analysis

Journal of Applied Probability ◽

10.2307/3215120 ◽

1995 ◽

Vol 32 (3) ◽

pp. 659-667 ◽

Cited By ~ 4

Author(s):

Michael C. Fu ◽

Jian-Qiang Hu

Keyword(s):

Perturbation Analysis ◽

Hazard Rate ◽

Rate Function ◽

Hazard Rates ◽

Hazard Rate Function ◽

Finite Support ◽

Restrictive Assumption ◽

Rate Functions

Many applications of smoothed perturbation analysis lead to estimators with hazard rate functions of underlying distributions. A key assumption used in proving unbiasedness of the resulting estimator is that the hazard rate function be bounded, a restrictive assumption which excludes all distributions with finite support. Here, we prove through a simple example that this assumption can in fact be removed.

Download Full-text

On unbounded hazard rates for smoothed perturbation analysis

Journal of Applied Probability ◽

10.1017/s0021900200103110 ◽

1995 ◽

Vol 32 (03) ◽

pp. 659-667 ◽

Cited By ~ 1

Author(s):

Michael C. Fu ◽

Jian-Qiang Hu

Keyword(s):

Perturbation Analysis ◽

Hazard Rate ◽

Rate Function ◽

Hazard Rates ◽

Hazard Rate Function ◽

Finite Support ◽

Restrictive Assumption ◽

Rate Functions

Many applications of smoothed perturbation analysis lead to estimators with hazard rate functions of underlying distributions. A key assumption used in proving unbiasedness of the resulting estimator is that the hazard rate function be bounded, a restrictive assumption which excludes all distributions with finite support. Here, we prove through a simple example that this assumption can in fact be removed.

Download Full-text

A monotonicity in reversible Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200001777 ◽

2006 ◽

Vol 43 (02) ◽

pp. 486-499 ◽

Cited By ~ 2

Author(s):

Robert Lund ◽

Ying Zhao ◽

Peter C. Kiessler

Keyword(s):

Convergence Rate ◽

Markov Chains ◽

State Space ◽

Hazard Rate ◽

Probability Generating Function ◽

Countable State Space ◽

Return Times ◽

Geometric Convergence ◽

Reversible Markov Chains ◽

Countable State

In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times toeverystate in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.

Download Full-text

Stability Analysis of the Nanjung Water Diversion Twin Tunnels based on Convergence Measurement

Indonesian Mining Professionals Journal ◽

10.36986/impj.v1i1.11 ◽

2019 ◽

Vol 1 (1) ◽

pp. 49-60

Author(s):

Simon Heru Prassetyo ◽

Ganda Marihot Simangunsong ◽

Ridho Kresna Wattimena ◽

Made Astawa Rai ◽

Irwandy Arif ◽

...

Keyword(s):

Stability Analysis ◽

Convergence Rate ◽

Critical Strain ◽

Convergence Rates ◽

Strain Analysis ◽

Water Diversion ◽

Tunnel Face ◽

Tunnel Stability ◽

The Stability ◽

Twin Tunnels

This paper focuses on the stability analysis of the Nanjung Water Diversion Twin Tunnels using convergence measurement. The Nanjung Tunnel is horseshoe-shaped in cross-section, 10.2 m x 9.2 m in dimension, and 230 m in length. The location of the tunnel is in Curug Jompong, Margaasih Subdistrict, Bandung. Convergence monitoring was done for 144 days between February 18 and July 11, 2019. The results of the convergence measurement were recorded and plotted into the curves of convergence vs. day and convergence vs. distance from tunnel face. From these plots, the continuity of the convergence and the convergence rate in the tunnel roof and wall were then analyzed. The convergence rates from each tunnel were also compared to empirical values to determine the level of tunnel stability. In general, the trend of convergence rate shows that the Nanjung Tunnel is stable without any indication of instability. Although there was a spike in the convergence rate at several STA in the measured span, that spike was not replicated by the convergence rate in the other measured spans and it was not continuous. The stability of the Nanjung Tunnel is also confirmed from the critical strain analysis, in which most of the STA measured have strain magnitudes located below the critical strain line and are less than 1%.

Download Full-text

Reinforcement learning versus swarm intelligence for autonomous multi-HAPS coordination

SN Applied Sciences ◽

10.1007/s42452-021-04658-6 ◽

2021 ◽

Vol 3 (6) ◽

Author(s):

Ogbonnaya Anicho ◽

Philip B. Charlesworth ◽

Gurvinder S. Baicher ◽

Atulya K. Nagar

Keyword(s):

Reinforcement Learning ◽

State Space ◽

Swarm Intelligence ◽

Performance Indicators ◽

Convergence Rates ◽

Tuning Parameters ◽

Continuous State Space ◽

Continuous State ◽

User Coverage ◽

Better Than

AbstractThis work analyses the performance of Reinforcement Learning (RL) versus Swarm Intelligence (SI) for coordinating multiple unmanned High Altitude Platform Stations (HAPS) for communications area coverage. It builds upon previous work which looked at various elements of both algorithms. The main aim of this paper is to address the continuous state-space challenge within this work by using partitioning to manage the high dimensionality problem. This enabled comparing the performance of the classical cases of both RL and SI establishing a baseline for future comparisons of improved versions. From previous work, SI was observed to perform better across various key performance indicators. However, after tuning parameters and empirically choosing suitable partitioning ratio for the RL state space, it was observed that the SI algorithm still maintained superior coordination capability by achieving higher mean overall user coverage (about 20% better than the RL algorithm), in addition to faster convergence rates. Though the RL technique showed better average peak user coverage, the unpredictable coverage dip was a key weakness, making SI a more suitable algorithm within the context of this work.

Download Full-text