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Product Brownian Measures

Ronald Pyke
Advances in Applied Probability
Vol. 18, Analytic and Geometric Stochastics: Papers in Honour of G. E. H. Reuter (Dec., 1986), pp. 117-131
Stable URL: http://www.jstor.org/stable/20528782
Page Count: 15
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Product Brownian Measures
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Abstract

If {Z₁(A): A ∈ 𝓐₁} and {Zβ‚‚(A): A ∈ 𝓐₂} are independent set-indexed processes, a product process Z may be defined on the set of rectangles 𝓐₁ Γ— 𝓐₂, by Z(A₁ Γ— Aβ‚‚) = Z₁(A₁)Zβ‚‚(Aβ‚‚). If each $Z_{i}$ is a continuous process with respect to some metric over $\scr{A}_{i}$, then this product process is also continuous relative to the natural product space metric. In this paper we assume that both Z₁ and Zβ‚‚ are Brownian processes and study the question of enlarging the domain of Z to as large a subfamily, 𝓐, of the product Οƒ-field, Οƒ(𝓐₁ Γ— 𝓐₂), as possible. It is shown that a continuous extension of Z to 𝓐 is possible under certain assumptions on 𝓐, the main one of which is that its log-entropy function, H, say, satisfies the integrability condition $\int_{o}^{1}H(u)u^{-\frac{1}{2}}du<\infty $. In order to prove that this extension is possible, exponential bounds are derived for the tail probabilities P[Z(A) > x] that are uniform for a sufficiently large sub-family of 𝓐.

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