Bernoulli Processes with Non-Positive Correlations

Abstract

We deal with sequences of dependent Bernoulli variables with non positive correlations. Some special models for 1-dependent and 2-dependent 0–1 valued variables are analyzed, namely Bernoulli variables obtained by clipping a linear combination of iid variables. Formulas describing dependence of their correlation function on the clipping parameters are derived. The lower bound for the sum of autocorrelations of Bernoulli variables obtained by clipping a Gaussian process is provided.

Keywords:

• Bernoulli variables
• Clipping
• Negatively dependent variables
• Two-block-factors
• 0–1 valued variables

Mathematics Subject Classification:

• 62H20
• 62M10
• 60K99

Acknowledgments

The work is a part of the research project MSM 0021620839 financed by MŠMT. We also gratefully acknowledge the support of the Grant Agency of Charles University in Prague (grant 56808) and the Grant Agency of the Czech Republic (grant 201/05/H007).

The referee is thanked for a careful inspection of the article and for his/her comments.