Baby wald

Let \(X=(X_i)_{i=1}^{\infty}\) be an iid sequence of random variables. Let \(N\) be a random nonnegative integer that is independent of \(X\). Suppose \(X_1\) and \(N\) have finite expectations. Prove that \[ \mathbb{E} \big( \sum_{i=1} ^N X_i \big) = \mathbb{E}N \mathbb{E} X_1\]

Solution

Observe that

\[\sum_{i=1} ^N X_i = \sum_{n=0} ^{\infty}\sum_{i=1} ^n X_i\mathbf{1}[N=n].\]

Note that we take as convention that when \(n=0\), we have the empty sum, which is zero. Since \(N\) and the iid sequence \(X\) are independent, we have that

\[\mathbb{E}(X_i \mathbf{1}[N=n] ) = \mathbb{E}(X_i) \mathbb{E}(\mathbf{1}[N=n] ) = \mathbb{E}(X_i) \mathbb{P}(N=n).\] Thus taking expectations on both sides (and bring the expectation operator over an infinite sum) we have

\[ \mathbb{E} \big( \sum_{i=1} ^N X_i \big) = \sum_{n=0} ^{\infty}\sum_{i=1} ^n \mathbb{E}(X_i) \mathbb{P}(N=n) = \mathbb{E}(X_1)\sum_{n=0}n\mathbb{P}(N=n) = \mathbb{E}(X_1) \mathbb{E}(N),\]

as desired.

Tiling

Solutions

  • Consider two tile types: a and b, of lengths \(a\) and \(b\), respectively. We can consider this an alternating renewal process: the working time is the time it remains in an a-tile, before switching to a b-tile, having service time, the time it remains in the b-tile. In order to compute the working time, we note that we spend time \(a\) on each time we get an a-tile; thus by the Baby Wald result the working time is given by \(aN_a\), where \(N_a\) is geometric with parameter \(p=\tfrac{1}{2}\). Similarly, the service time is given by \(bN_b\), and again \(N_b\) is just geometric with parameter \(p=\tfrac{1}{2}\). Applying the theorem on alternating renewal processes we have

\[\lim_{t \to \infty} \mathbb{P}(\text{that we are on an a-tile at time } t) = \frac{\mathbb{E}( aN_a) }{\mathbb{E}( aN_a) + \mathbb{E}( bN_b) } = \frac{a}{a+b}\]

  • Consider the case of three tiles \(a\), \(b\), and \(c\) that occur with probabilities \(p_a + p_b + p_c =1\). Again, we can treat this as an alternating renewal process. The working time is \(aN_a\), where \(N_a\) is now a geometric \(1-p_a\). The service time is a bit more complicated, as remaining on a b-tile or a c-tile counts as service; the number of such tiles is again geometric \(1-p_b-p_c = p_a\), but we really do need to use the Baby Wald, as the tiles could be or length \(b\) or \(c\), with probabilites \(\tfrac{p_b}{p_b+p_c}\) and \(\tfrac{p_c}{p_b+p_c}\) respectively, giving an expecting service time of

\[\big( b \cdot \tfrac{p_b}{p_b+p_c} + c \cdot \tfrac{p_c}{p_b+p_c} \big) \cdot \tfrac{1}{p_a}.\]

Thus

\[\lim_{t \to \infty} \mathbb{P}(\text{that we are on an a-tile at time } t) = \frac{\tfrac{a}{1-p_a}} { \tfrac{a}{1-p_a} + (b \cdot \tfrac{p_b}{p_b+p_c} + c \cdot \tfrac{p_c}{p_b+p_c} \big) \cdot \tfrac{1}{p_a} }.\]

  • We test this formula in Python, with \(a=1, b=\sqrt{2}, c=\pi\) and \(p_a = 2/9\), \(p_b=3/9\), and \(p_c = 4/9\). We check the tile type at \(t=300\), for a total of \(1000\) independent times.
import numpy as np

pa= 2/9
pb= 3/9
pc= 4/9

a=1
b=np.sqrt(2)
c=np.pi

def type():
  x=a
  u=np.random.uniform()
  if (u > pa and u < pa+pb):
    x=b
  if (u > pa+pb):
    x=c
  return x

def tile(t):
  ctile = type()
  tlength = ctile
  while(t > tlength):
    ctile = type()
    tlength = tlength + ctile
  return ctile

y = [tile(300) for _ in range(1000)  ]

freq = np.unique(y,return_counts = True)

print(freq)
## (array([1.        , 1.41421356, 3.14159265]), array([ 90, 229, 681], dtype=int64))
tile1 = (1/1000)*freq[1][0]


tile1theory =   (a/(1-pa)) / (    a/(1-pa)    +  (b*(pb/(pb+pc)) +c *(pc/(pb + pc)))*(1/pa)                   ) 
print(tile1 - tile1theory      )
## -0.016332011180342706

Size-biased intervals

Let \(\Pi\) be a Poisson point process on \([0, \infty)\). Pick a (large) number, say \(x=\sqrt{2} + 100\). Find the smallest interval \((A,B)\) such that \(x \in (A,B)\), and \(A\) and \(B\) are points of \(\Pi\), thus \(B\) is the next arrival after \(A\).

Solutions

Consider a Poisson process of intensity \(\lambda\)

  • We already know from the memoryless property that \(B-x\) is still exponentially distributed with rate \(\lambda\). It is not hard to argue from reversibility that \(x-A\) is also exponentially distributed with rate \(\lambda\).

One can also think about generating a two-sided Poisson process on \((-\infty, -\infty)\), this can be accomplished by putting together two independent Poisson processes on \((0, \infty)\) glued at \(0\), where one is flipped.

  • We see that \(S\) has actually the sum of two independent exponentials.

  • We can also see this in the following Python code, where we take \(\lambda\)=1:


# Poisson arrivals just past the end time

def pois(end):
  T= np.array([np.random.exponential()])
  while(T[-1]<end):
    T = np.append(T, T[-1] + np.random.exponential())
  return T

def find(times,x):
  n=0
  while(x > times[n]):
    n = n+1
  return (times[n] - times[n-1])

print(find(pois(200), 100+np.sqrt(2) ))

## 1.8857454328843772
y = [find(pois(200), 100+np.sqrt(2) ) for _ in range(5000)  ]

print(np.mean(y))
## 1.9517397077124503
import matplotlib.pyplot as plt
supress=plt.hist(y, bins = 100, density=True, label='Proability Histogram') 
t=np.linspace(0,10,num=1000)
plt.plot(t,t*np.exp(-1*t), label='Exact Density')
plt.legend(loc='upper left')
plt.show()

Endnotes