First passage problems for upwards skip-free random walks via the Φ,W,Z\Phi,W,Z paradigm

We develop the theory of the $W$ and $Z$ scale functions for right-continuous (upwards skip-free) discrete-time discrete-space random walks, along the lines of the analogue theory for spectrally negative L\'evy processes. Notably, we introduce for the first time in this context the one and two-parameter scale functions $Z$, which appear for example in the joint problem of deficit at ruin and time of ruin, and in problems concerning the walk reflected at an upper barrier. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous. The theory is shown to be fruitful by providing a convenient unified framework for studying dividends-capital injection problems under various objectives, for the so-called compound binomial risk model of actuarial science.Comment: 27 page

Non-Gaussian halo abundances in the excursion set approach with correlated steps

We study the effects of primordial non-Gaussianity on the large scale structure in the excursion set approach, accounting for correlations between steps of the random walks in the smoothed initial density field. These correlations are induced by realistic smoothing filters (as opposed to a filter that is sharp in k-space), but have been ignored by many analyses to date. We present analytical arguments -- building on existing results for Gaussian initial conditions -- which suggest that the effect of the filter at large smoothing scales is remarkably simple, and is in fact identical to what happens in the Gaussian case: the non-Gaussian walks behave as if they were smooth and deterministic, or "completely correlated". As a result, the first crossing distribution (which determines, e.g., halo abundances) follows from the single-scale statistics of the non-Gaussian density field -- the so-called "cloud-in-cloud" problem does not exist for completely correlated walks. Also, the answer from single-scale statistics is simply one half that for sharp-k walks. We explicitly test these arguments using Monte Carlo simulations of non-Gaussian walks, showing that the resulting first crossing distributions, and in particular the factor 1/2 argument, are remarkably insensitive to variations in the power spectrum and the defining non-Gaussian process. We also use our Monte Carlo walks to test some of the existing prescriptions for the non-Gaussian first crossing distribution. Since the factor 1/2 holds for both Gaussian and non-Gaussian initial conditions, it provides a theoretical motivation (the first, to our knowledge) for the common practice of analytically prescribing a ratio of non-Gaussian to Gaussian halo abundances.Comment: 11 pages, 7 figures; v2 -- fixed a formatting problem + typos; v3 -- minor changes, accepted in MNRA

Scaling limits for random processes from the point of view of group cohomology (Women in Mathematics)

Based on joint work with Kenichi Bannai and Yukio KametaniScaling limits for random walks and stochastic interacting systems have been intensively studied for decades and still remain one of the very major themes of probability theory. I have mainly worked on the diffusive scaling limits in space-time for interacting particle systems, in which I have recently revealed the importance of a group cohomological view and its connection with the Hodge decomposition as well as periodic matrices with collaborators. This structure is expected to be common and effective not only for interacting particle systems but also in the case of one-particle random walks and in homogenization problems. I would like to introduce this connection between the scale limits for random processes and the group cohomology