Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 4 / b

Table of contents
- Solution b

Solution

Assume the avoidance formula. Given another admissible hull

C

, put

D = A \cup ψ_{A}^{- 1} (C)

with the bounded filling. Uniqueness of the normalized maps gives

ψ_{D} = ψ_{C} \circ ψ_{A}, ψ_{D}^{'} (0) = ψ_{C}^{'} (0) ψ_{A}^{'} (0) .

Therefore

P (ψ_{A} (γ) \cap C = \emptyset ∣ γ \cap A = \emptyset) = \frac{P ( γ \cap D = \emptyset )}{P ( γ \cap A = \emptyset )} = \frac{ψ _{D}^{'} ( 0 ) ^{α}}{ψ _{A}^{'} ( 0 ) ^{α}} = ψ_{C}^{'} (0)^{α} .

These avoidance events determine the law of a simple closed random set. They agree with those of

γ

, so the conditional mapped law equals the original law. Hence the avoidance formula implies the chordal restriction property.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version