Consider a Lévy process
X
t
X_t
with quadratic variation process
V
t
=
σ
2
t
+
∑
0
>
s
≤
t
(
Δ
X
s
)
2
V_t=\sigma ^2 t+ \sum _{0>s\le t} (\Delta X_s)^2
,
t
>
0
t>0
, where
Δ
X
t
=
X
t
−
X
t
−
\Delta X_t=X_t-X_{t-}
denotes the jump process of
X
X
. We give stability and compactness results, as
t
↓
0
t\,\downarrow \,0
, for the convergence both of the deterministically normed (and possibly centered) processes
X
t
X_t
and
V
t
V_t
, as well as theorems concerning the “self-normalised” process
X
t
/
V
t
X_{t}/\sqrt {V_t}
. Thus, we consider the stochastic compactness and convergence in distribution of the 2-vector
(
(
X
t
−
a
(
t
)
)
/
b
(
t
)
,
V
t
/
b
(
t
)
)
\left ((X_t-a(t))/b(t), V_t/b(t)\right )
, for deterministic functions
a
(
t
)
a(t)
and
b
(
t
)
>
0
b(t)>0
, as
t
↓
0
t\,\downarrow \,0
, possibly through a subsequence; and the stochastic compactness and convergence in distribution of
X
t
/
V
t
X_{t}/\sqrt {V_t}
, possibly to a nonzero constant (for stability), as
t
↓
0
t\,\downarrow \,0
, again possibly through a subsequence.
As a main application it is shown that
X
t
/
V
t
⟶
D
N
(
0
,
1
)
X_{t}/\sqrt {V_t}\stackrel {\mathrm {D}}{\longrightarrow } N(0,1)
, a standard normal random variable, as
t
↓
0
t\,\downarrow \,0
, if and only if
X
t
/
b
(
t
)
⟶
D
N
(
0
,
1
)
X_t/b(t)\stackrel {\mathrm {D}}{\longrightarrow } N(0,1)
, as
t
↓
0
t\downarrow 0
, for some nonstochastic function
b
(
t
)
>
0
b(t)>0
; thus,
X
t
X_t
is in the domain of attraction of the normal distribution, as
t
↓
0
t\,\downarrow \,0
, with or without centering constants being necessary (these being equivalent).
We cite simple analytic equivalences for the above properties, in terms of the Lévy measure of
X
X
. Functional versions of the convergences are also given.