| 1. |
- Renlund, Henrik, 1979-
(author)
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First-passage percolation on ladder-like graphs with heterogeneous exponential times.
- 2011
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Reports (other academic/artistic)abstract
- We determine the asymptotic speed of the first-passage percolation process on some ladder-like graphs (or width-2 stretches) when the times associated with different edges are independent and exponentially distributed but not necessarily all with the same mean. The method uses a particular Markov chain associated with the first-passage percolation process and properties of its stationary distribution.
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| 2. |
- Renlund, Henrik, 1979-
(author)
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Limit theorems for stochastic approximation algorithms.
- 2011
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Reports (other academic/artistic)abstract
- We prove a central limit theorem applicable to one dimensional stochastic approximation algorithms that converge to a point where the error terms of the algorithm do not vanish. We show how this applies to a certain class of these algorithms that in particular covers a generalized Pólya urn model, which is also discussed. In addition, we show how to scale these algorithms in some cases where we cannot determine the limiting distribution but expect it to be non-normal.
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| 3. |
- Renlund, Henrik, 1979-
(author)
-
Recursive Methods in Urn Models and First-Passage Percolation
- 2011
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Doctoral thesis (other academic/artistic)abstract
- This PhD thesis consists of a summary and four papers which deal with stochastic approximation algorithms and first-passage percolation. Paper I deals with the a.s. limiting properties of bounded stochastic approximation algorithms in relation to the equilibrium points of the drift function. Applications are given to some generalized Pólya urn processes. Paper II continues the work of Paper I and investigates under what circumstances one gets asymptotic normality from a properly scaled algorithm. The algorithms are shown to converge in some other circumstances, although the limiting distribution is not identified. Paper III deals with the asymptotic speed of first-passage percolation on a graph called the ladder when the times associated to the edges are independent, exponentially distributed with the same intensity. Paper IV generalizes the work of Paper III in allowing more edges in the graph as well as not having all intensities equal.
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