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The longest increasing subsequence (LIS) of a sequence of correlated randomvariables is a basic quantity with potential applications that has started toreceive proper attention only recently. Here we investigate the behavior of thelength of the LIS of the so-called symmetric ultra-fat tailed random walk,introduced earlier in an abstract setting in the mathematical literature. Afterexplicit constructing the ultra-fat tailed random walk, we found numericallythat the expected length $L_{n}$ of its LIS scales with the length $n$ of thewalk like $\langle L_{n} \rangle \sim n^{0.716}$, indicating that, indeed, asfar as the behavior of the LIS is concerned the ultra-fat tailed distributioncan be thought of as equivalent to a very heavy tailed $\alpha$-stabledistribution. We also found that the distribution of $L_{n}$ seems to beuniversal, in agreement with results obtained for other heavy tailed randomwalks.

A numerical investigation into the scaling behavior of the longest increasing subsequences of the symmetric ultra-fat tailed random walk