281 research outputs found
On Layered Stable Processes
Layered stable (multivariate) distributions and processes are defined and
studied. A layered stable process combines stable trends of two different
indices, one of them possibly Gaussian. More precisely, in short time, it is
close to a stable process while, in long time, it approximates another stable
(possibly Gaussian) process. We also investigate the absolute continuity of a
layered stable process with respect to its short time limiting stable process.
A series representation of layered stable processes is derived, giving insights
into both the structure of the sample paths and of the short and long time
behaviors. This series is further used for sample paths simulation.Comment: 22 pages, 9 figure
On Fractional Tempered Stable Motion
Fractional tempered stable motion (fTSm)} is defined and studied. FTSm has
the same covariance structure as fractional Brownian motion, while having tails
heavier than Gaussian but lighter than stable. Moreover, in short time it is
close to fractional stable L\'evy motion, while it is approximately fractional
Brownian motion in long time. A series representation of fTSm is derived and
used for simulation and to study some of its sample path properties.Comment: 25 pages, 6 figure
Sparse Long Blocks and the Micro-Structure of the Longest Common Subsequences
Consider two random strings having the same length and generated by an iid
sequence taking its values uniformly in a fixed finite alphabet. Artificially
place a long constant block into one of the strings, where a constant block is
a contiguous substring consisting only of one type of symbol. The long block
replaces a segment of equal size and its length is smaller than the length of
the strings, but larger than its square-root. We show that for sufficiently
long strings the optimal alignment corresponding to a Longest Common
Subsequence (LCS) treats the inserted block very differently depending on the
size of the alphabet. For two-letter alphabets, the long constant block gets
mainly aligned with the same symbol from the other string, while for three or
more letters the opposite is true and the block gets mainly aligned with gaps.
We further provide simulation results on the proportion of gaps in blocks of
various lengths. In our simulations, the blocks are "regular blocks" in an iid
sequence, and are not artificially inserted. Nonetheless, we observe for these
natural blocks a phenomenon similar to the one shown in case of
artificially-inserted blocks: with two letters, the long blocks get aligned
with a smaller proportion of gaps; for three or more letters, the opposite is
true.
It thus appears that the microscopic nature of two-letter optimal alignments
and three-letter optimal alignments are entirely different from each other.Comment: To appear: Journal of Statistical Physic
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