Strict Sense Stationary Signal

A strict-sense stationary process is one whose complete statistical structure does not change with time — all joint distributions remain identical under time shifts.

For SSS, the following all must be time-invariant:

• Mean

• Variance

• Autocorrelation

• All higher-order moments

• All joint PDFs

• All marginal PDFs

Everything.

That’s why SSS is a very strong condition.

A random process $X(t)$ is strict-sense stationary if for every:

• Positive integer $n$

• Any set of time instants $t_1, t_2, \dots, t_n$

• Any time shift $\tau$

the joint distribution satisfies:

$$F_{X(t_1),\dots ,X(t_n)}(x_1,\dots ,x_n)=F_{X(t_1+\tau ),\dots ,X(t_n+\tau )}(x_1,\dots ,x_n)$$

A Large Strictly Stationary Example

Let’s define a random process where:

X(t) = 0 with probability 1/2

X(t) = 1 with probability 1/2

Each time point is independent. This is like an infinite sequence of fair coin flips.

t:    1  2  3  4  5  6  7  8  9  10 ...

X(t): 1  0  1  1  0  0  1  0  1   1 ...

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Joint Probability of Many Values

Pick 5 time points: X(10), X(11), X(12), X(13), X(14)

We want:

P(X(10)=1, X(11)=0, X(12)=1, X(13)=1, X(14)=0)

Because each is independent with probability 1/2:

P = (1/2) × (1/2) × (1/2) × (1/2) × (1/2)

  = 1/32

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Shift by Any Amount h

Shift by h = 20. Now consider:

X(30), X(31), X(32), X(33), X(34)

We want the same pattern:

P(X(30)=1, X(31)=0, X(32)=1, X(33)=1, X(34)=0) = 1/32

The joint probability did not change after shifting. This is strict stationarity.

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Large Example: 100 Time Points

The probability of ANY specific pattern of 100 values is:

P = (1/2)^100

After shifting time by any h:

P = (1/2)^100

Again, the joint probability stays the same → strict stationarity.

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Is Joint Probability Just Multiplication?

Yes — ONLY when the variables are independent

P(A ∩ B ∩ C) = P(A) × P(B) × P(C)

In our fair-coin process:

P(X(t1)=x1, ..., X(tn)=xn) = Π P(X(ti)=xi)

No — if variables are dependent

Example: if X(t+1) = X(t) always, then:

P(X(1)=0, X(2)=0) = P(X(1)=0)

You cannot multiply because the variables are not independent.

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Final Summary

• Independence → joint probability = multiplication of each probability
• Dependence → cannot multiply
• In the fair-coin process:
• each time has same distribution
• all times independent
• shifting does not change joint probability
• → strictly stationary

Further Reading

1. Wide Sense Stationary Signal