Basic properties
The unconditional probability density function at a fixed time t:
The expectation is zero:
- E(Wt) = 0.
The variance is t:
The covariance and correlation:
The results for the expectation and variance follow immediately from the definition that increments have a normal distribution, centred at zero. Thus
The results for the covariance and correlation follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that t1 < t2.
Substitute the simple identity :
Since W(t1) = W(t1) − W(t0) and W(t2) − W(t1), are independent,
Thus
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