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Introduction to Stochastic Processes

Module by: Tuan Do-Hong

Definitions, distributions, and stationarity

Definition 1: Stochastic Process
Given a sample space, a stochastic process is an indexed collection of random variables defined for each ω∈Ω ω Ω .
∀t,t∈ℝ: X t ω t t X t ω (1)
Example 1 
Received signal at an antenna as in Figure 1.
Figure 1
For a given tt, X t ω X t ω is a random variable with a distribution
First-order distribution F X t b=Pr X t ≤b=Pr{ω∈Ω| X t ω≤b} F X t b X t b ω Ω X t ω b (2)
Definition 2: First-order stationary process
If F X t b F X t b is not a function of time then X t X t is called a first-order stationary process.
Second-order distribution F X t 1 , X t 2 b 1 b 2 =Pr X t 1 ≤ b 1 X t 2 ≤ b 2 F X t 1 , X t 2 b 1 b 2 X t 1 b 1 X t 2 b 2 (3)
for all t 1 ∈ℝ t 1 , t 2 ∈ℝ t 2 , b 1 ∈ℝ b 1 , b 2 ∈ℝ b 2
Nth-order distribution F X t 1 , X t 2 , … , X t N b 1 b 2 … b N =Pr X t 1 ≤ b 1 … X t N ≤ b N F X t 1 , X t 2 , … , X t N b 1 b 2 … b N X t 1 b 1 … X t N b N (4)
NNth-order stationary : A random process is stationary of order NN if
F X t 1 , X t 2 , … , X t N b 1 b 2 … b N = F X t 1 + T , X t 2 + T , … , X t N + T b 1 b 2 … b N F X t 1 , X t 2 , … , X t N b 1 b 2 … b N F X t 1 + T , X t 2 + T , … , X t N + T b 1 b 2 … b N (5)
Strictly stationary : A process is strictly stationary if it is NNth order stationary for all NN.
Example 2 
X t =cos2π f 0 t+Θω X t 2 f 0 t Θ ω where f 0 f 0 is the deterministic carrier frequency and Θω : Ω→ℝ Θ ω : → Ω is a random variable defined over -ππ and is assumed to be a uniform random variable; i.e., f Θ θ=12πifθ∈-ππ0otherwise f Θ θ 1 2 θ 0
F X t b=Pr X t ≤b=Prcos2π f 0 t+Θ≤b F X t b X t b 2 f 0 t Θ b (6)
F X t b=Pr-π≤2π f 0 t+Θ≤-arccosb+Prarccosb≤2π f 0 t+Θ≤π F X t b 2 f 0 t Θ b b 2 f 0 t Θ (7)
F X t b=∫-π-2π f 0 t-arccosb-2π f 0 t12πdθ+∫arccosb-2π f 0 tπ-2π f 0 t12πdθ=2π-2arccosb12π F X t b θ 2 f 0 t b 2 f 0 t 1 2 θ b 2 f 0 t 2 f 0 t 1 2 2 2 b 1 2 (8)
f X t x=ddx1-1πarccosx=1π1-x2if|x|≤10otherwise f X t x x 1 1 x 1 1 x 2 x 1 0 (9)
This process is stationary of order 1.
Figure 2
The second order stationarity can be determined by first considering conditional densities and the joint density. Recall that
X t =cos2π f 0 t+Θ X t 2 f 0 t Θ (10)
Then the relevant step is to find
Pr X t 2 ≤ b 2 | X t 1 = x 1 X t 1 x 1 X t 2 b 2 (11)
Note that
X t 1 = x 1 =cos2π f 0 t+Θ⇒Θ=arccos x 1 -2π f 0 t X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t (12)
X t 2 =cos2π f 0 t 2 +arccos x 1 -2π f 0 t 1 =cos2π f 0 t 2 - t 1 +arccos x 1 X t 2 2 f 0 t 2 x 1 2 f 0 t 1 2 f 0 t 2 t 1 x 1 (13)
Figure 3
F X t 2 , X t 1 b 2 b 1 =∫-∞ b 1 f X t 1 x 1 Pr X t 2 ≤ b 2 | X t 1 = x 1 d x 1 F X t 2 , X t 1 b 2 b 1 x 1 b 1 f X t 1 x 1 X t 1 x 1 X t 2 b 2 (14)
Note that this is only a function of t 2 - t 1 t 2 t 1 .
Example 3 
Every TT seconds, a fair coin is tossed. If heads, then X t =1 X t 1 for nT≤t<n+1T n T t n 1 T . If tails, then X t =-1 X t -1 for nT≤t<n+1T n T t n 1 T .
Figure 4
p X t x=12ifx=112ifx=-1 p X t x 1 2 x 1 1 2 x -1 (15)
for all t∈ℝ t . X t X t is stationary of order 1.
Second order probability mass function
p X t 1 X t 2 x 1 x 2 = p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1 p X t 1 X t 2 x 1 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 1 x 1 (16)
The conditional pmf
p X t 2 | X t 1 x 2 | x 1 =0if x 2 ≠ x 1 1if x 2 = x 1 p X t 2 | X t 1 x 2 | x 1 0 x 2 x 1 1 x 2 x 1 (17)
when nT≤ t 1 <n+1T n T t 1 n 1 T and nT≤ t 2 <n+1T n T t 2 n 1 T for some nn.
p X t 2 | X t 1 x 2 | x 1 = p X t 2 x 2 p X t 2 | X t 1 x 2 | x 1 p X t 2 x 2 (18)
for all x 1 x 1 and for all x 2 x 2 when nT≤ t 1 <n+1T n T t 1 n 1 T and mT≤ t 2 <m+1T m T t 2 m 1 T with n≠m n m
p X t 2 X t 1 x 2 x 1 =0if x 2 ≠ x 1 for  nT≤ t 1 , t 2 <n+1T p X t 1 x 1 if x 2 = x 1 for  nT≤ t 1 , t 2 <n+1T p X t 1 x 1 p X t 2 x 2 if n≠m for  nT≤ t 1 <n+1T∧mT≤ t 2 <m+1T p X t 2 X t 1 x 2 x 1 0 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 x 2 x 1 for  n T t 1 , t 2 n 1 T p X t 1 x 1 p X t 2 x 2 n m for  n T t 1 n 1 T m T t 2 m 1 T (19)

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