## Law of large numbers for monotone convolution

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##### Date

2014-09-19##### Author

Wendler, Enzo

##### Type

Thesis##### Degree Level

Masters##### Abstract

In this thesis, we use martingales to show that the dilation of a sequence of monotone convolutions $D_\frac{1}{b_n} (\mu_1 \triangleright \mu_2 \triangleright \cdots \triangleright \mu_n)$ is stable, where $\mu_j$ are probability distributions with the condition $\sum \limits_{n=1}^\infty \frac{1}{b_n} \text{var}(\mu_n) < \infty$. This proves a law of large numbers for monotonically independent random variables.

##### Degree

Master of Science (M.Sc.)##### Department

Mathematics and Statistics##### Program

Mathematics##### Supervisor

Wang, Jiun-Chau##### Committee

Srinivasan, Raj; Samei, Ebrahim; Dutchyn, Christopher##### Copyright Date

September 2014##### Subject

Law of large numbers

monotone convolution

Non-commutative probability theory

Markov chains and martingales