Strong consistency and other properties of the spectral variance estimator

Consistent estimation of the variance parameter of a stochastic process allows construction, under certain conditions, of a confidence interval for the mean of the process. If the variance estimator is strongly consistent, fixed-width confidence interval construction is valid for large samples. It has long been known that the spectral variance estimator of steady-state simulation output analysis is consistent in the mean-square sense. Here, we provide strong consistency of this estimator, thereby justifying fixed-width confidence interval construction for the spectral method. A characterization of spectral density function estimators is also introduced. This characterization provides insight into the relation between spectral methods and overlapping batch means-type variance estimators. Finally, some of the mathematical conditions provide qualitative insight into the relation between the process correlation and certain parameters of spectral methods.

[1]  J. Doob.  Stochastic processes , 1953 .

[2]  P. Whittle,et al.  Bounds for the Moments of Linear and Quadratic Forms in Independent Variables , 1960 .

[3]  H. Robbins,et al.  ON THE ASYMPTOTIC THEORY OF FIXED-WIDTH SEQUENTIAL CONFIDENCE INTERVALS FOR THE MEAN. , 1965 .

[4]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[5]  P. Major,et al.  An approximation of partial sums of independent RV'-s, and the sample DF. I , 1975 .

[6]  W. Philipp,et al.  Almost sure invariance principles for partial sums of weakly dependent random variables , 1975 .

[7]  E. Nummelin,et al.  A splitting technique for Harris recurrent Markov chains , 1978 .

[8]  K. Athreya,et al.  A New Approach to the Limit Theory of Recurrent Markov Chains , 1978 .

[9]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[10]  P. Révész,et al.  Strong approximations in probability and statistics , 1981 .

[11]  P. Glynn.  Some New Results in Regenerative Process Theory. , 1982 .

[12]  Lee W. Schruben,et al.  Confidence Interval Estimation Using Standardized Time Series , 1983, Oper. Res..

[13]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[14]  Paul Bratley,et al.  A guide to simulation , 1983 .

[15]  Bruce W. Schmeiser,et al.  Overlapping batch means: something for nothing? , 1984, WSC '84.

[16]  P. Glynn,et al.  A new class of strongly consistent variance estimators for steady-state simulations , 1988 .

[17]  李幼升,et al.  Ph , 1989 .

[18]  Donald L. Iglehart,et al.  Simulation Output Analysis Using Standardized Time Series , 1990, Math. Oper. Res..

[19]  P. Glynn,et al.  The Asymptotic Validity of Sequential Stopping Rules for Stochastic Simulations , 1992 .