I want to get simulation observation with kernel Density, but I have the Following error type: TypeError: unsupported operand type(s) for +: 'float' and 'NoneType'

how to resolve it?
This is the code that I use:

 from matplotlib.pyplot import *
 from math import *
 from array import *
 import numpy as np 
 from numpy.random import *
 from scipy.misc import *
 from scipy.stats import *
 from scipy import *
 from random import *


 N=1000
 n=30
 lamb=0.5
 X=lamb*tan(pi*(np.reshape(rand(n,1),n)-0.5)) #loi de Cauchy 
 x=1
 alpha=0.45

 def k_gaussien(x):
 sigma=1
 if(sigma<=0):
 return((1/(sigma*sqrt(2*pi)))*exp(-(x**2/(2*sigma**2))))

 def h(n,alpha):
 for i in range(1,n): 
 return (i**(1-alpha))

 def f_PR(x,X,alpha):
 global F;
 F = ones((n,1))
 h_n = h(n,alpha)
 for k in range(2,n):
 for i in range(1,k):
 F[k] = F[k-1] + k_gaussien((x-X[i])*i**alpha)
 F[k] = F[k-1] * h_n
 return F

 # Almost sure convergence f_n(x)--> f(x) ps
 figure(figsize=(20,10))
 fPR=f_PR(x,X,alpha)
 T=linspace(1,n,n);
 plot(cumsum(fPR)/T)
 plot(T,(1/pi)*(lamb/(lamb**2 + y**2))*linspace(1,1,N),"r--",lw=3)#with Cauchy density
 grid(True)
 title("convergence presque sure",fontsize=20,color="blue")

 #Convergence in mean N(0,e2f(x))
 hist(fPR,bins=linspace(-10,10,50),normed=True)
 y=linspace(-10,10,100);
 v=(1/pi)*(lamb/(lamb**2 + y**2))
 plot(y,(1/sqrt(2*pi)*v)*exp ((-(y*y)/(2*v**2)))*linspace(1,1,n),'r--') #with cauchy density
 title("convergence asymptotique", fontsize=20,color="blue")

TypeError Traceback (most recent call last)

<ipython-input-76-13bc86608417> in <module>()
 38 # Almost sure convergence f_n(x)--> f(x) ps
 39 figure(figsize=(20,10))
 ---> 40 fPR=f_PR(x,X,alpha)
 41 T=linspace(1,n,n);
 42 plot(cumsum(fPR)/T)

 <ipython-input-76-13bc86608417> in f_PR(x, X, alpha)
 32 for k in range(2,n):
 33 for i in range(1,k):
 ---> 34 F[k] = F[k-1] + k_gaussien((x-X[i])*i**alpha)
 35 F[k] = F[k-1] * h_n
 36 return F

TypeError: unsupported operand type(s) for +: 'float' and 'NoneType'

It's in your definition:

def k_gaussien(x):
 sigma=1
 if(sigma<=0):
 return((1/(sigma*sqrt(2*pi)))*exp(-(x**2/(2*sigma**2))))

You hardcoded sigma=1, but your function only returns something if sigma<=0 which will never be the case. So k_gaussien((x-X[i])*i**alpha) will return None. Therefore F[k] = F[k-1] + k_gaussien((x-X[i])*i**alpha) tries to sum float and None types, which does not work.