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Consider a continuum of random variables such that each takes the value $1$ with probability $p$ and $0$ with probability $1-p$. The random variables should be essentially pairwise independent.

Sun and Zhang (2009, Journal of Economic Theory, 144, p. 432-443, doi:10.1016/j.jet.2008.05.001, Theorem 1 and Corollary 2) show that there exists a $\Omega$, a suitable probability space on $\Omega$, an extension $\bar \lambda$ of the Lebesgue measure on $[0,1]$, a Fubini extension on $[0,1]\times\Omega$ and a corresponding process $g\colon [0,1]\times \Omega \mapsto \mathbf R$, such that the random variables $g(t, \cdot)$ are essentially pairwise independent with the required distribution and a law of large numbers is valid. I hope this correctly summarizes their findings, but I am grateful for any corrections. Now my question is how to formulate a law of large numbers here. I don't want to use their formulation as it is too general for me and it requires even more notation.

Is the statement "$\int_{[0,1]} g(t,\omega) \mathrm{d}\bar \lambda=\bar \lambda(\{t\in [0,1]|g(t,\omega)=1\})=p\ \ $ almost surely" correct?

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