H*Edgeworth expansion for Stirling numbers of the first kind
We use the expansion and the notation from the paper
"General Edgeworth expansions with applications to profiles of random trees"
by Zakhar Kabluchko, Alexander Marynych, Henning Sulzbach
available at http:arxiv.orgabs1606.03920
*L
In[4]:=
Out[4]=
Out[5]=
Out[6]=
H*Parameters:
phiHbetaL = Exp@betaD-1,
LogW@betaD = LogGamma@thetaD-LogGamma@theta*Exp@betaDD,
kappa@@jDD = j-th derivative of phiHbetaL=Exp@betaD-1 at beta=0,
chi@@jDD = j-th derivative of LogW at beta=0,
sigma@@jDD = 1 is ignored everywhere.
*L
kappa = Table@1, 8n, 1, 5<D
LogW@beta_D = LogGamma@thetaD - LogGamma@theta * Exp@betaDD
chi = List@D@LogW@xD, xD . x ® 0,
D@D@LogW@xD, xD, xD . x ® 0, D@D@D@LogW@xD, xD, xD, xD . x ® 0D
81, 1, 1, 1, 1<
LogGamma@thetaD - LogGamma@ãbeta thetaD
9- theta PolyGamma@0, thetaD,
- theta PolyGamma@0, thetaD - theta2 PolyGamma@1, thetaD,
- theta PolyGamma@0, thetaD 3 theta2 PolyGamma@1, thetaD - theta3 PolyGamma@2, thetaD=
In[7]:=
H*Differential operators D_1,D_2,D_3. Here D stays for sigma^H-1L ddx *L
D1 = kappa@@3DD  6 * D ^ 3 + chi@@1DD * D ^ 1
D2 = kappa@@4DD  12 * D ^ 4 + chi@@2DD * D ^ 2
D3 = kappa@@5DD  20 * D ^ 5 + chi@@3DD * D ^ 3
D3
Out[7]=
- D theta PolyGamma@0, thetaD
6
D4
Out[8]=
12
D5
Out[9]=
20
+ D2 I- theta PolyGamma@0, thetaD - theta2 PolyGamma@1, thetaDM
+ D3 I- theta PolyGamma@0, thetaD -
3 theta2 PolyGamma@1, thetaD - theta3 PolyGamma@2, thetaDM
2
stirling.nb
In[10]:=
H*
Bell polynomials are
B_0=1,
B_1Hz_1L = z_1,
B_2Hz_1,z_2L = z_1^2 + z_2,
B_3Hz_1,z_2,z_3L = z_1^3 + 3z_1z_2 + z_3
*L
H*
Probabilist Hermite polynomials are given by
He_pHxL = 2^H-p2L*HermiteH@p, xSqrt@2DD
To check this,
use: Table@Simplify@2^H-p2L*HermiteH@p, xSqrt@2DDD, 8p,0,6<D
*L
H*Terms in the Edgeworth expansion of the
Stirling numbers are denoted by G_0HxL, G_1HxL,...*L
H0 = 1;
In[11]:=
H1@x_D = Simplify@ HExpand@D1D . D ^ p_ -> 2 ^ H- p  2L * HermiteH@p, x  Sqrt@2DDL .
D -> 2 ^ H- 1  2L * HermiteH@1, x  Sqrt@2DDD
x I- 3 + x2 - 6 theta PolyGamma@0, thetaDM
1
Out[11]=
6
In[27]:=
A11 = Simplify@Coefficient@H1@xD, xDD
A12 = Simplify@Coefficient@H1@xD, x ^ 3DD
1
Out[27]=
-
- theta PolyGamma@0, thetaD
2
1
Out[28]=
6
In[12]:=
H2@x_D = Simplify@
1  H2 !L * Expand@D1 ^ 2 + D2D . D ^ p_ -> 2 ^ H- p  2L * HermiteH@p, x  Sqrt@2DDD
1
Out[12]=
72
In[32]:=
I- 6 + 27 x2 - 12 x4 + x6 - 12 theta x2 I- 3 + x2 M PolyGamma@0, thetaD +
36 theta2 I- 1 + x2 M PolyGamma@0, thetaD2 - 36 theta2 I- 1 + x2 M PolyGamma@1, thetaDM
A21 = Simplify@H2@0DD
A22 = Simplify@Coefficient@H2@xD, x ^ 2DD
1
Out[32]=
12
1
Out[33]=
8
I- 1 - 6 theta2 PolyGamma@0, thetaD2 + 6 theta2 PolyGamma@1, thetaDM
I3 + 4 theta PolyGamma@0, thetaD +
4 theta2 PolyGamma@0, thetaD2 - 4 theta2 PolyGamma@1, thetaDM
stirling.nb
In[15]:=
H3@x_D = Simplify@1  H3 !L * Expand@D1 ^ 3 + 3 D1 * D2 + D3D .
D ^ p_ -> 2 ^ H- p  2L * HermiteH@p, x  Sqrt@2DDD
1
Out[15]=
6480
x I810 - 2115 x2 + 999 x4 - 135 x6 + 5 x8 + 540 theta2 I- 3 - 4 x2 + x4 M PolyGamma@0, thetaD2 1080 theta3 I- 3 + x2 M PolyGamma@0, thetaD3 -
540 theta2 I- 3 - 4 x2 + x4 M PolyGamma@1, thetaD + 90 theta PolyGamma@0, thetaD
I6 - 27 x2 + 12 x4 - x6 + 36 theta2 I- 3 + x2 M PolyGamma@1, thetaDM +
3240 theta3 PolyGamma@2, thetaD - 1080 theta3 x2 PolyGamma@2, thetaDM
In[34]:=
A31 = Simplify@Coefficient@H3@xD, xDD
1
Out[34]=
24
I3 - 6 theta2 PolyGamma@0, thetaD2 + 12 theta3 PolyGamma@0, thetaD3 +
6 theta2 PolyGamma@1, thetaD + PolyGamma@0, thetaD
I2 theta - 36 theta3 PolyGamma@1, thetaDM + 12 theta3 PolyGamma@2, thetaDM
H*
The first three terms in the expansion for x = aSqrt@wD.
The term H4w^2 = constantw^2 +
oH1w^2L is ignored because the constant does not depend on a.
Error: oH1w^2L
*L
Series@Exp@- t ^ 2  2D, 8t, 0, 4<D
t2
Out[14]=
1-
t4
+
2
8
+ O@tD5
3
4
stirling.nb
In[16]:=
A = H1 + H1@a  Sqrt@wDD  Sqrt@wD + H2@a  Sqrt@wDD  w + H3@a  Sqrt@wDD  w ^ H3  2LL *
H1 - a ^ 2  H2 wL + 1  8 * a ^ 4  w ^ 2L;
Collect@Coefficient@A, 1  wD, aD
Collect@Coefficient@A, 1  w ^ H1  2LD, aD
Collect@Coefficient@A, 1  w ^ H3  2LD, aD
Collect@Coefficient@A, 1  w ^ 2D, aD
a2
1
Out[17]=
-
-
12
1
-
2
theta2 PolyGamma@0, thetaD2 +
2
1
a -
1
- theta PolyGamma@0, thetaD +
2
Out[18]=
0
Out[19]=
0
a4
+ a3
Out[20]=
8
a2
a
In[21]:=
theta2 PolyGamma@1, thetaD
2
5
1
+
theta PolyGamma@0, thetaD +
12 2
5
1
3
+ theta PolyGamma@0, thetaD + theta2 PolyGamma@0, thetaD2 12 2
4
3
theta2 PolyGamma@1, thetaD +
4
1
1
1
+
theta PolyGamma@0, thetaD - theta2 PolyGamma@0, thetaD2 +
8 12
4
1
1
3
theta3 PolyGamma@0, thetaD3 + theta2 PolyGamma@1, thetaD - theta3
2
4
2
1
PolyGamma@0, thetaD PolyGamma@1, thetaD + theta3 PolyGamma@2, thetaD
2
H*Here we insert a = gam + chi@@1DD-12, where gam is the new variable*L
Poly@gam_D = Simplify@Coefficient@A, 1  w ^ 2D . a ® gam + chi@@1DD - 1  2D
H- 1 + 2 gam - 2 theta PolyGamma@0, thetaDL
384
I1 + 18 gam + 44 gam2 + 24 gam3 + 4 H- 19 + 6 gamL theta2 PolyGamma@0, thetaD2 +
1
Out[21]=
24 theta3 PolyGamma@0, thetaD3 - 24 H- 5 + 6 gamL theta2 PolyGamma@1, thetaD 2 theta PolyGamma@0, thetaD I13 + 44 gam - 12 gam2 + 72 theta2 PolyGamma@1, thetaDM +
96 theta3 PolyGamma@2, thetaDM
In[22]:=
H*
Now we can compute the quantity denoted by s^*
HthetaL in the paper "Asymptotic expansions for the Ewens
distribution and the Stirling numbers of the first kind"
*L
Simplify@Poly@1  2D - Poly@- 1  2DD
1
Out[22]=
2
theta2 H2 PolyGamma@1, thetaD + theta PolyGamma@2, thetaDL
stirling.nb
In[24]:=
H*The resulting expression is positive*L
Plot@Poly@1  2D - Poly@- 1  2D, 8theta, 0, 1<D
0.4
0.3
Out[24]=
0.2
0.1
0.2
0.4
0.6
5