Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
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Equivalence-Singularity Dichotomies of Gaussian and Poisson Processes from The Kolmogorov's Zero-One Law

  • Park, Jeong-Soo (Department of Statistics, Chonnam National University, 300 Yongbond-dong, Buk-gu)
  • Published : 1994.12.01
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Abstract

Let P and Q be probability measures of a measurable space $(\Omega, F)$, and ${F_n}_{n \geq 1}$ be a sequence of increasing sub $\sigma$-fields which generates F. For each $n \geq 1$, let $P_n$ and $Q_n$ be the restrictions of P and Q to $F_n$, respectively. Under the assumption that $Q_n \ll P_n$ for every $n \geq 1$, a zero-one condition is derived for P and Q to have the dichotomy, i.e., either $Q \ll P$ or $Q \perp P$. Then using this condition and the Kolmogorov's zero-one law, we give new and simple proofs of the dichotomy theorems for a pair of Gaussian measures and Poisson processes with examples.

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