# When are touchpoints limits for generalized Pólya urns

@inproceedings{Pemantle1991WhenAT, title={When are touchpoints limits for generalized P{\'o}lya urns}, author={Robin Pemantle}, year={1991} }

Hill, Lane, and Sudderth (1980) consider a Polya-like urn scheme in which X 0 , X 1 ,..., are the successive proportions of red balls in an urn to which at the n th stage a red ball is added with probability f(X n ) and a black ball is added with probability 1−f(X n ). For continuous f they show that X n converges almost surely to a random limit X which is a fixed point for f and ask whether the point p can be a limit if p is a touchpoint, i.e. p=f(p) but f(x)>x for x¬=;p in a neighborhood of p

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## References

SHOWING 1-3 OF 3 REFERENCES

### Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations

- Mathematics
- 1990

A particle in Rd moves in discrete time. The size of the nth step is of order 1/n and when the particle is at a position v the expectation of the next step is in the direction F(v) for some fixed…