Local fluctuations of critical
Mandelbrot cascades
Konrad Kolesko
joint with
D. Buraczewski and P. Dyszewski
Warwick, 18-22 May, 2015
Random measures
µ
µ1
µ2
For given random variables X1 , X2 s.t. E e−X1 + e−X2 = 1 we
are interested in random measures µ on [0, 1) satisfying self
similar property:
µ(B) = e−X1 µ1 2(B ∩ [0, 1/2)) + e−X2 µ2 2(B ∩ [1/2, 1) − 1) ,
where µ1 ⊥⊥ µ2 ⊥⊥ X1 , X2 and Lµ = Lµ1 = Lµ2 .
Goal: Understand local properties of µ
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
2 / 19
Random measures
µ
µ1
µ2
For given random variables X1 , X2 s.t. E e−X1 + e−X2 = 1 we
are interested in random measures µ on [0, 1) satisfying self
similar property:
µ(B) = e−X1 µ1 2(B ∩ [0, 1/2)) + e−X2 µ2 2(B ∩ [1/2, 1) − 1) ,
where µ1 ⊥⊥ µ2 ⊥⊥ X1 , X2 and Lµ = Lµ1 = Lµ2 .
Goal: Understand local properties of µ
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
2 / 19
Random measures
µ
µ1
µ2
For given random variables X1 , X2 s.t. E e−X1 + e−X2 = 1 we
are interested in random measures µ on [0, 1) satisfying self
similar property:
µ(B) = e−X1 µ1 2(B ∩ [0, 1/2)) + e−X2 µ2 2(B ∩ [1/2, 1) − 1) ,
where µ1 ⊥⊥ µ2 ⊥⊥ X1 , X2 and Lµ = Lµ1 = Lµ2 .
Goal: Understand local properties of µ
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
2 / 19
Phase transition
Define ψ(t) := log2 E e−tX1 + e−tX2 .
By assumption ψ(0) = 1, ψ(1) = 0, ψ % ∞.
Supercritical
Critical
Subcritical
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
3 / 19
Subcritical case ψ 0 (1) = −m < 0
Theorem (Liu)
Suppose that ψ 0 (1) = −m < 0, then for any δ > 0, almost all
realization of µ, µ-almost all x and sufficiently large n
√
2
µ(B(x, 2−n )) ≥ 2−nm−(1+δ) 2σ n log log n
√
2
µ(B(x, 2−n )) ≤ 2−nm+(1+δ) 2σ n log log n ,
where σ 2 = ψ 00 (1) − ψ(1). Moreover
µ(B(x, 2−n )) ≤ 2−nm−(1−δ)
µ(B(x, 2−n )) ≥ 2−nm+(1−δ)
Konrad Kolesko
√
2σ 2 n log log n
√
2σ 2 n log log n
Local fluctuations of critical Mandelbrot cascades
i.o.
i.o.
18-22 May, 2015
4 / 19
Subcritical case ψ 0 (1) = −m < 0
Theorem (Liu)
Suppose that ψ 0 (1) = −m < 0, then for any δ > 0, almost all
realization of µ, µ-almost all x and sufficiently large n
√
2
µ(B(x, 2−n )) ≥ 2−nm−(1+δ) 2σ n log log n
√
2
µ(B(x, 2−n )) ≤ 2−nm+(1+δ) 2σ n log log n ,
where σ 2 = ψ 00 (1) − ψ(1). Moreover
µ(B(x, 2−n )) ≤ 2−nm−(1−δ)
µ(B(x, 2−n )) ≥ 2−nm+(1−δ)
Konrad Kolesko
√
2σ 2 n log log n
√
2σ 2 n log log n
Local fluctuations of critical Mandelbrot cascades
i.o.
i.o.
18-22 May, 2015
4 / 19
Supercritical & critical case
Barral, Rhodes, Vargas
When ψ 0 (1) > 0 then µ is purely atomic
Barral, Kupiainen, Nikula, Saksman, Webb
If ψ 0 (1) = 0, then the random measure µ almost surely has no
atoms. Moreover for any k and δ > 0, µ-a.e. x
√
√
− 6 log 2 n(log n+(1/3+δ) log log n)
−n
for
µ(B(x, 2 )) ≥ e
sufficiently large n
√
√
µ(B(x, 2−n )) ≥ e−( 2 log 2+δ) n log n i.o.
µ(B(x, 2−n )) ≤ n−k for sufficiently large n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
5 / 19
Supercritical & critical case
Barral, Rhodes, Vargas
When ψ 0 (1) > 0 then µ is purely atomic
Barral, Kupiainen, Nikula, Saksman, Webb
If ψ 0 (1) = 0, then the random measure µ almost surely has no
atoms. Moreover for any k and δ > 0, µ-a.e. x
√
√
− 6 log 2 n(log n+(1/3+δ) log log n)
−n
for
µ(B(x, 2 )) ≥ e
sufficiently large n
√
√
µ(B(x, 2−n )) ≥ e−( 2 log 2+δ) n log n i.o.
µ(B(x, 2−n )) ≤ n−k for sufficiently large n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
5 / 19
Supercritical & critical case
Barral, Rhodes, Vargas
When ψ 0 (1) > 0 then µ is purely atomic
Barral, Kupiainen, Nikula, Saksman, Webb
If ψ 0 (1) = 0, then the random measure µ almost surely has no
atoms. Moreover for any k and δ > 0, µ-a.e. x
√
√
− 6 log 2 n(log n+(1/3+δ) log log n)
−n
for
µ(B(x, 2 )) ≥ e
sufficiently large n
√
√
µ(B(x, 2−n )) ≥ e−( 2 log 2+δ) n log n i.o.
µ(B(x, 2−n )) ≤ n−k for sufficiently large n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
5 / 19
Supercritical & critical case
Barral, Rhodes, Vargas
When ψ 0 (1) > 0 then µ is purely atomic
Barral, Kupiainen, Nikula, Saksman, Webb
If ψ 0 (1) = 0, then the random measure µ almost surely has no
atoms. Moreover for any k and δ > 0, µ-a.e. x
√
√
− 6 log 2 n(log n+(1/3+δ) log log n)
−n
for
µ(B(x, 2 )) ≥ e
sufficiently large n
√
√
µ(B(x, 2−n )) ≥ e−( 2 log 2+δ) n log n i.o.
µ(B(x, 2−n )) ≤ n−k for sufficiently large n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
5 / 19
Main theorem
Theorem (D. Buraczewski, P. Dyszewski, K.K.)
Let ψ 0 (1) = 0, k ∈ N and δ > 0. Then for almost all realizations
of µ, µ-almost all x ∈ [0, 1) and sufficiently large n we have
p
µ(B(x, 2−n )) ≥ exp −(1 + δ) 2σ 2 n log log n
!
√
−
n
µ(B(x, 2−n )) ≤ exp Qk
2
i=1 log(i) n (log(k+1) n)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
6 / 19
Main theorem
Theorem (D. Buraczewski, P. Dyszewski, K.K.)
Let ψ 0 (1) = 0, k ∈ N and δ > 0. Then for almost all realizations
of µ, µ-almost all x ∈ [0, 1) and sufficiently large n we have
p
µ(B(x, 2−n )) ≥ exp −(1 + δ) 2σ 2 n log log n
!
√
n
−
µ(B(x, 2−n )) ≤ exp Qk
2
i=1 log(i) n (log(k+1) n)
e−K1
√
n(log n+(1/3+δ) log log n)
√
e−(K2 +δ)
Konrad Kolesko
n−k
n log n
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
6 / 19
Notations
Dyadic intervals on [0,1) ! vertices of a binary tree T
x ∈ [0, 1) ! θ ∈ ∂T .
B(v) = {θ ∈ ∂T : v ∈ oθ}.
For a random measure µω we are interested in a pointwise
estimates of µω (B(θn )) on the enlarged measure space
e
P(dω,
dθ) := P(dω)µω (dθ)
i.e. we are looking for deterministic φ1 , φ2 s.t.
φ1 (n) ≤ µω (B(θn )) ≤ φ2 (n),
e
for P-almost
all (ω, θ) and large n.
e can be replaced by P(dω,
b
P
dθ) = P(dω)µω (dθ), where µω is the
normalized measure µω .
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
7 / 19
Notations
Dyadic intervals on [0,1) ! vertices of a binary tree T
x ∈ [0, 1) ! θ ∈ ∂T .
B(v) = {θ ∈ ∂T : v ∈ oθ}.
For a random measure µω we are interested in a pointwise
estimates of µω (B(θn )) on the enlarged measure space
e
P(dω,
dθ) := P(dω)µω (dθ)
i.e. we are looking for deterministic φ1 , φ2 s.t.
φ1 (n) ≤ µω (B(θn )) ≤ φ2 (n),
e
for P-almost
all (ω, θ) and large n.
e can be replaced by P(dω,
b
P
dθ) = P(dω)µω (dθ), where µω is the
normalized measure µω .
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
7 / 19
Notations
Dyadic intervals on [0,1) ! vertices of a binary tree T
x ∈ [0, 1) ! θ ∈ ∂T .
B(v) = {θ ∈ ∂T : v ∈ oθ}.
For a random measure µω we are interested in a pointwise
estimates of µω (B(θn )) on the enlarged measure space
e
P(dω,
dθ) := P(dω)µω (dθ)
i.e. we are looking for deterministic φ1 , φ2 s.t.
φ1 (n) ≤ µω (B(θn )) ≤ φ2 (n),
e
for P-almost
all (ω, θ) and large n.
e can be replaced by P(dω,
b
P
dθ) = P(dω)µω (dθ), where µω is the
normalized measure µω .
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
7 / 19
Branching random walk
o
P – probability measure on the set of labeled trees
X(v) – the sum of random variables along the path between o
and v
(X(u) = X2 + X21 + X212 )
{X(v)}v∈T – Branching random walk (BRW)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
8 / 19
Branching random walk
o
X1
X2
P – probability measure on the set of labeled trees
X(v) – the sum of random variables along the path between o
and v
(X(u) = X2 + X21 + X212 )
{X(v)}v∈T – Branching random walk (BRW)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
8 / 19
Branching random walk
o
X1
X11
X2
X12
P – probability measure on the set of labeled trees
X(v) – the sum of random variables along the path between o
and v
(X(u) = X2 + X21 + X212 )
{X(v)}v∈T – Branching random walk (BRW)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
8 / 19
Branching random walk
o
X1
X11
X2
X12
X21
X22
P – probability measure on the set of labeled trees
X(v) – the sum of random variables along the path between o
and v
(X(u) = X2 + X21 + X212 )
{X(v)}v∈T – Branching random walk (BRW)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
8 / 19
Branching random walk
o
X1
X11
X2
X12
X21
X22
X212
u
P – probability measure on the set of labeled trees
X(v) – the sum of random variables along the path between o
and v
(X(u) = X2 + X21 + X212 )
{X(v)}v∈T – Branching random walk (BRW)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
8 / 19
Branching random walk
o
X1
X11
X2
X12
X21
X22
X212
u
P – probability measure on the set of labeled trees
X(v) – the sum of random variables along the path between o
and v
(X(u) = X2 + X21 + X212 )
{X(v)}v∈T – Branching random walk (BRW)
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
8 / 19
Stopping line
P
P
Since E |v|=1 e−X(v) = 1 and E |v|=1 X(v)e−X(v) = 0 the
equation
X
Ef (Y ) := E
f (X(v))e−X(v)
|v|=1
defines distribution of a driftless r.v. Y
Let h be a harmonic function on some set A (a solution of a
Dirichlet problem), Vn = Y1 + · · · + Yn and
σ = min{k : s + Vk ∈
/ A}. Then the process h(s + Vmin(n,σ) ) is
a martingale.
Let τ = {w : s + X(w) ∈
/ A for the first time}, vτ = min(v, τ ).
Then
X
Wns =
h(s + X(vτ ))e−X(vτ )
is a martingale.
Konrad Kolesko
|v|=n
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
9 / 19
Stopping line
P
P
Since E |v|=1 e−X(v) = 1 and E |v|=1 X(v)e−X(v) = 0 the
equation
X
Ef (Y ) := E
f (X(v))e−X(v)
|v|=1
defines distribution of a driftless r.v. Y
Let h be a harmonic function on some set A (a solution of a
Dirichlet problem), Vn = Y1 + · · · + Yn and
σ = min{k : s + Vk ∈
/ A}. Then the process h(s + Vmin(n,σ) ) is
a martingale.
Let τ = {w : s + X(w) ∈
/ A for the first time}, vτ = min(v, τ ).
Then
X
Wns =
h(s + X(vτ ))e−X(vτ )
is a martingale.
Konrad Kolesko
|v|=n
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
9 / 19
Stopping line
P
P
Since E |v|=1 e−X(v) = 1 and E |v|=1 X(v)e−X(v) = 0 the
equation
X
Ef (Y ) := E
f (X(v))e−X(v)
|v|=1
defines distribution of a driftless r.v. Y
Let h be a harmonic function on some set A (a solution of a
Dirichlet problem), Vn = Y1 + · · · + Yn and
σ = min{k : s + Vk ∈
/ A}. Then the process h(s + Vmin(n,σ) ) is
a martingale.
Let τ = {w : s + X(w) ∈
/ A for the first time}, vτ = min(v, τ ).
Then
X
Wns =
h(s + X(vτ ))e−X(vτ )
is a martingale.
Konrad Kolesko
|v|=n
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
9 / 19
Martingales
Some natural martingales
h ≡ 1: Wn =
P
h(x) = x: Dn =
|v|=n
e−X(v)
P
|v|=n
X(v)e−X(v)
A = [0,P
∞), h(x) ≈ x ∨ 0:
Wns = |v|=n h(s + X(v))1[s+X(v0 )>0, for v0 ≤v] e−X(v)
µ(B(v)) := lim
n
Konrad Kolesko
X
X(w)e−X(w)
|w|=n, w<v
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
10 / 19
Martingales
Some natural martingales
h ≡ 1: Wn =
P
h(x) = x: Dn =
|v|=n
e−X(v) → 0
P
|v|=n
X(v)e−X(v) → D > 0
A = [0,P
∞), h(x) ≈ x ∨ 0:
Wns = |v|=n h(s + X(v))1[s+X(v0 )>0, for v0 ≤v] e−X(v)
µ(B(v)) := lim
n
Konrad Kolesko
X
X(w)e−X(w)
|w|=n, w<v
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
10 / 19
Martingales
Some natural martingales
h ≡ 1: Wn =
P
h(x) = x: Dn =
|v|=n
e−X(v) → 0
P
|v|=n
X(v)e−X(v) → D > 0
A = [0,P
∞), h(x) ≈ x ∨ 0:
Wns = |v|=n h(s + X(v))1[s+X(v0 )>0, for v0 ≤v] e−X(v)
µ(B(v)) := lim
n
Konrad Kolesko
X
X(w)e−X(w)
|w|=n, w<v
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
10 / 19
Martingales
Some natural martingales
h ≡ 1: Wn =
P
h(x) = x: Dn =
|v|=n
e−X(v) → 0
P
|v|=n
X(v)e−X(v) → D > 0
A = [0,P
∞), h(x) ≈ x ∨ 0:
Wns = |v|=n h(s + X(v))1[s+X(v0 )>0, for v0 ≤v] e−X(v)
µ(B(v)) := lim
n
Konrad Kolesko
X
X(w)e−X(w)
|w|=n, w<v
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
10 / 19
Wns
Wns =
X
h(s + X(v))1[s+X(v0 )>0, for v0 ≤v] e−X(v)
|v|=n
Wns → W s
P-a.s. and L1
For any s ∈ D define
Ps :=
Ws
·P
h(s)
P[supp W s ] & 1 − 1/s
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
11 / 19
Ps
=⇒ BRW
∞
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
12 / 19
Ps
BRW
∞
=⇒
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
12 / 19
Ps
BRW
∞
=⇒
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
12 / 19
Spinal decomposition of Ps
BRW
∞
s
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
s
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
X11
X21
s
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
X11
1
e−X1 h(s + X11 )
X21
1
e−X2 h(s + X21 )
s
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S1 = s+X21
S0 = s
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
X12
S1
X22
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
X12
2
e−X1 h(S1 + X12 )
S1
X22
2
e−X2 h(S1 + X22 )
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S1
S2 = S1 +X21
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S1
S2
X13
X23
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S1
S2
X13
e
−X13
h(S2 +
X13 )
X23
−X23
e
S0
h(S2 +
X23 )
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S1
S3 = S2 +X31
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
Random tree Ts
with a distinguished
ray Θ ∈ ∂Ts
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
s
s
P(T ∈ dω) = P (dω)
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
Spinal decomposition of Ps
bs (dω, dθ) :=
P
P(Ts ∈ dω, Θ ∈ dθ)
BRW
∞
S5
S4
S1
S3
S2
S0
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
13 / 19
bs vs. P
b
P
We have
bs P.
b
P
The converse is not true, but for any set A
bs (A) = 1 ⇒ P(A)
b
P
≥ 1 − (s),
for some (s) → 0 as s → ∞. In particular, if for any s > 0
bs (A) = 1
P
then
b
P(A)
= 1.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
14 / 19
Probabilistic interpretation of local fluctuations
e−(S0 −s)
∞
C0
e−(S1 −s)
S1
S3
S2
C1
R0
S5
S4
e−(S2 −s)
C2
S0
R1
R2
0
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
15 / 19
Probabilistic interpretation of local fluctuations
bs the sequence µω (B(θn )) has the same law as
Under P
X
e−(Sk −s) Ck Rk ,
k≥n
where Sk is a random walk starting form s conditioned to stay
positive, Rk independent random variables.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
15 / 19
Useful information
We are looking for LIL for
P
k≥n
e−Sk +s Ck Rk ,
Hambly, Kersting, Kyprianou
√
ψ(t)
dt < ∞ iff Sn > nψ(n)
t
lim supn √ 2Sn
=1
2nσ log log n
R∞
√
eventually
p
nψ(n) < Sn < (1 + δ) 2nσ 2 log log n
Ps -a.s. for n sufficiently large
Kyprianou
1 − Ee−tR ∼ tL(t) near 0 =⇒ ERγ < ∞ for γ < 1
In particular, by Borel-Cantelli lemma, Rn < n2 for all but
finitely many n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
16 / 19
Useful information
We are looking for LIL for
P
k≥n
e−Sk Rk ,
Hambly, Kersting, Kyprianou
√
ψ(t)
dt < ∞ iff Sn > nψ(n)
t
lim supn √ 2Sn
=1
2nσ log log n
R∞
√
eventually
p
nψ(n) < Sn < (1 + δ) 2nσ 2 log log n
Ps -a.s. for n sufficiently large
Kyprianou
1 − Ee−tR ∼ tL(t) near 0 =⇒ ERγ < ∞ for γ < 1
In particular, by Borel-Cantelli lemma, Rn < n2 for all but
finitely many n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
16 / 19
Useful information
We are looking for LIL for
P
k≥n
e−Sk Rk ,
Hambly, Kersting, Kyprianou
√
ψ(t)
dt < ∞ iff Sn > nψ(n)
t
lim supn √ 2Sn
=1
2nσ log log n
R∞
√
eventually
p
nψ(n) < Sn < (1 + δ) 2nσ 2 log log n
Ps -a.s. for n sufficiently large
Kyprianou
1 − Ee−tR ∼ tL(t) near 0 =⇒ ERγ < ∞ for γ < 1
In particular, by Borel-Cantelli lemma, Rn < n2 for all but
finitely many n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
16 / 19
Useful information
We are looking for LIL for
P
k≥n
e−Sk Rk ,
Hambly, Kersting, Kyprianou
√
ψ(t)
dt < ∞ iff Sn > nψ(n)
t
lim supn √ 2Sn
=1
2nσ log log n
R∞
√
eventually
p
nψ(n) < Sn < (1 + δ) 2nσ 2 log log n
Ps -a.s. for n sufficiently large
Kyprianou
1 − Ee−tR ∼ tL(t) near 0 =⇒ ERγ < ∞ for γ < 1
In particular, by Borel-Cantelli lemma, Rn < n2 for all but
finitely many n.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
16 / 19
Upper bound
Take any ψ such that
R∞
ψ(t)dt
t
X
< ∞. We have that
e−Sk Rk
k≥n
is eventually bounded by
X
e−
√
kψ(k) 2
k .
k≥n
On the other hand, if
X
R∞
ψ(t)dt
t
= ∞ then
i.o.
√
e−Sk Rk ≥ e−Sn Rn ≥ e−
nψ(n)
i.o.
Rn ≥ δe−
√
nψ(n)
k≥n
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
17 / 19
Upper bound
Take any ψ such that
R∞
ψ(t)dt
t
X
< ∞. We have that
e−Sk Rk
k≥n
is eventually bounded by
X
√
e−
kψ(k)
.
k≥n
On the other hand, if
X
R∞
ψ(t)dt
t
= ∞ then
i.o.
√
e−Sk Rk ≥ e−Sn Rn ≥ e−
nψ(n)
i.o.
Rn ≥ δe−
√
nψ(n)
k≥n
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
17 / 19
Upper bound
Take any ψ such that
R∞
ψ(t)dt
t
X
< ∞. We have that
e−Sk Rk
k≥n
is eventually bounded by
X
e−k · polynomial(k).
√
k≥ nψ(n)
On the other hand, if
X
R∞
ψ(t)dt
t
= ∞ then
i.o.
√
e−Sk Rk ≥ e−Sn Rn ≥ e−
nψ(n)
i.o.
Rn ≥ δe−
√
nψ(n)
k≥n
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
17 / 19
Upper bound
Take any ψ such that
R∞
ψ(t)dt
t
X
< ∞. We have that
e−Sk Rk
k≥n
is eventually bounded by
X
e−k .
√
k≥ nψ(n)
On the other hand, if
X
R∞
ψ(t)dt
t
= ∞ then
i.o.
√
e−Sk Rk ≥ e−Sn Rn ≥ e−
nψ(n)
i.o.
Rn ≥ δe−
√
nψ(n)
k≥n
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
17 / 19
Upper bound
Take any ψ such that
R∞
ψ(t)dt
t
X
< ∞. We have that
e−Sk Rk
k≥n
is eventually bounded by
√
e−
On the other hand, if
X
R∞
ψ(t)dt
t
nψ(n)
.
= ∞ then
i.o.
√
e−Sk Rk ≥ e−Sn Rn ≥ e−
nψ(n)
i.o.
Rn ≥ δe−
√
nψ(n)
k≥n
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
17 / 19
Upper bound
Take any ψ such that
R∞
ψ(t)dt
t
X
< ∞. We have that
e−Sk Rk
k≥n
is eventually bounded by
√
e−
On the other hand, if
X
R∞
ψ(t)dt
t
nψ(n)
.
= ∞ then
i.o.
√
e−Sk Rk ≥ e−Sn Rn ≥ e−
nψ(n)
i.o.
Rn ≥ δe−
√
nψ(n)
k≥n
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
17 / 19
Lower bound
Take q > 1 and by N (n) := q dlogq ne . Borel-Cantelli’s lemma
implies that for sufficiently large n, sup Rk ≥ δ0 .
q n <k≤q n+1
X
e−Sk Rk ≥
k≥n
X
e−(1+δ)
√
2kσ 2 log log k
Rk
k≥n
qN (n)
≥
X
√
2
e−(1+δ) 2kσ log log k Rk
k=N (n)
√
2
≥ δ0 e−(1+δ) 2qN (n)σ log log(qN (n))
√ 2 2
2
≥ δ0 e−(1+δ) 2q nσ log log(q n)
√
2
≥ e−(1+2δ) 2nσ log log n ,
for some q.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
18 / 19
Lower bound
Take q > 1 and by N (n) := q dlogq ne . Borel-Cantelli’s lemma
implies that for sufficiently large n, sup Rk ≥ δ0 .
q n <k≤q n+1
X
e−Sk Rk ≥
k≥n
X
e−(1+δ)
√
2kσ 2 log log k
Rk
k≥n
qN (n)
≥
X
√
2
e−(1+δ) 2kσ log log k Rk
k=N (n)
√
2
≥ δ0 e−(1+δ) 2qN (n)σ log log(qN (n))
√ 2 2
2
≥ δ0 e−(1+δ) 2q nσ log log(q n)
√
2
≥ e−(1+2δ) 2nσ log log n ,
for some q.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
18 / 19
Lower bound
Take q > 1 and by N (n) := q dlogq ne . Borel-Cantelli’s lemma
implies that for sufficiently large n, sup Rk ≥ δ0 .
q n <k≤q n+1
X
e−Sk Rk ≥
k≥n
X
e−(1+δ)
√
2kσ 2 log log k
Rk
k≥n
qN (n)
≥
X
√
2
e−(1+δ) 2kσ log log k Rk
k=N (n)
√
2
≥ δ0 e−(1+δ) 2qN (n)σ log log(qN (n))
√ 2 2
2
≥ δ0 e−(1+δ) 2q nσ log log(q n)
√
2
≥ e−(1+2δ) 2nσ log log n ,
for some q.
Konrad Kolesko
Local fluctuations of critical Mandelbrot cascades
18-22 May, 2015
18 / 19