define white noise interms of brownian motion with examples in detail?

Answer 1

2. White noise as the derivative of a Brownian motion
White noise can be thought of as the derivative of a Brownian motion. But what is a Brownian motion? As is well-known, Robert Brown made microscopic observations in 1827 that small particles contained in the pollen of plants, when immersed in a liquid, exhibit highly irregular motions. This highly irregular motion is called Brownian motion. Mathematically, a Brownian motion is a continuous stationary stochastic process B(t) having independent increments and for each t, B(t) is a Gaussian random variable with mean 0 and variance t. It can be shown that B(t) is nowhere differentiable, a mathematical fact explaining the highly irregular motions that Robert Brown observed. This means that white noise, being thought of as the derivative dB(t)/dt of B(t), does not exist in the ordinary sense.