Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Distributional representations and dominance of a L\'{e}vy process over
  its maximal jump processes   Distributional identities for a L\'evy process $X_t$, its quadratic variation
process $V_t$ and its maximal jump processes, are derived, and used to make
"small time" (as $t\downarrow0$) asymptotic comparisons between them. The
representations are constructed using properties of the underlying Poisson
point process of the jumps of $X$. Apart from providing insight into the
connections between $X$, $V$, and their maximal jump processes, they enable
investigation of a great variety of limiting behaviours. As an application, we
study "self-normalised" versions of $X_t$, that is, $X_t$ after division by
$\sup_{0<s\le t}\Delta X_s$, or by $\sup_{0<s\le t}| \Delta X_s|$. Thus, we
obtain necessary and sufficient conditions for $X_t/\sup_{0<s\le t}\Delta X_s$
and $X_t/\sup_{0<s\le t}| \Delta X_s|$ to converge in probability to 1, or to
$\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates,
its largest jump. The former situation tends to occur when the singularity at 0
of the L\'evy measure of $X$ is fairly mild (its tail is slowly varying at 0),
while the latter situation is related to the relative stability or attraction
to normality of $X$ at 0 (a steeper singularity at 0). An important component
in the analyses is the way the largest positive and negative jumps interact
with each other. Analogous "large time" (as $t\to \infty$) versions of the
results can also be obtained.

36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
1
The categories of this paper are: cs
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom