Let
K
\mathbb {K}
be a field and let
S
=
K
[
x
1
,
…
,
x
n
]
S=\mathbb {K}[x_1,\dots ,x_n]
be the polynomial ring in
n
n
variables over
K
\mathbb {K}
. Assume that
I
⊂
S
I\subset S
is a squarefree monomial ideal. For every integer
k
≥
1
k\geq 1
, we denote the
k
k
-th symbolic power of
I
I
by
I
(
k
)
I^{(k)}
. Recently, Montaño and Núñez-Betancourt (2018), and independently Nguyen and Trung (to appear), proved that for every pair of integers
k
,
i
≥
1
k, i\geq 1
,
d
e
p
t
h
(
S
/
I
(
k
)
)
≤
d
e
p
t
h
(
S
/
I
(
⌈
k
i
⌉
)
)
.
\begin{equation*} \mathrm {depth}(S/I^{(k)})\leq \mathrm {depth}(S/I^{(\lceil \frac {k}{i}\rceil )}). \end{equation*}
We provide an alternative proof for this inequality. Moreover, we re-prove the known results that the sequence
{
d
e
p
t
h
(
S
/
I
(
k
)
)
}
k
=
1
∞
\{\mathrm {depth}(S/I^{(k)})\}_{k=1}^{\infty }
is convergent and
min
k
d
e
p
t
h
(
S
/
I
(
k
)
)
=
lim
k
→
∞
d
e
p
t
h
(
S
/
I
(
k
)
)
=
n
−
ℓ
s
(
I
)
,
\begin{equation*} \min _k\mathrm {depth}(S/I^{(k)})=\lim _{k\rightarrow \infty }\mathrm {depth}(S/I^{(k)})=n-\ell _s(I), \end{equation*}
where
ℓ
s
(
I
)
\ell _s(I)
denotes the symbolic analytic spread of
I
I
. We also determine an upper bound for the index of depth stability of symbolic powers of
I
I
. Next, we consider the Stanley depth of symbolic powers and prove that the sequences
{
s
d
e
p
t
h
(
S
/
I
(
k
)
)
}
k
=
1
∞
\{\mathrm {sdepth}(S/I^{(k)})\}_{k=1}^{\infty }
and
{
s
d
e
p
t
h
(
I
(
k
)
)
}
k
=
1
∞
\{\mathrm {sdepth}(I^{(k)})\}_{k=1}^{\infty }
are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.