Hi, Page 20-Chapter 2 White noise is sequence of iid random variables. I have 2 doubts 1. As random variables are independent, I think it should have independent increments. difference between 2 random variables will not depend on past. 2. For white noise the markov property holds. However, Random variables are independent, therefore, future value is independent of current & past values. Then how markov property is satisfied. I also feel other way round also. If markov property is satisfited for any process than process also has independent increment as Morkov is independent of past so increment will be also independent of past. Regards, Ashok
Hi Ashok, 1. No. Let Zn = toss a coin, 1 for heads, -1 for tails (a white noise process) Let Xn = Zn - Zn-1 (difference in two coin tosses) What is the probability that X2 = 2? 0.25, right? Because you would need Z2= heads, Z1 = tails Now, let me tell you that X1 = 2. Now, what is the probability that X2 = 2? 0, because it is impossible that Z1 = tails. So, X2 is NOT independent of X1. ie Zn does NOT have independent increments. Learn this example, it's a good example of a process that has the Markov property but does NOT have independent increments. (This answers your last point too). 2. I think you have misunderstood here what Markov means. Markov is NOT saying that future values must depend on the current value. What it is saying is that future values depend on NO MORE than the current value. White Noise is trivially Markov... Let's say we know that Z2 = 1 What is the probability that Z3 = 1? 0.5. (the value of Z2 is irrelevant) Ah, but what if I told you that Z1 = -1 and Z0 = -1? Also, irrelevant - knowing the past values does not change our opinions of what might happen in the future. Knowing the current value of Z2 is enough. In fact, we don't even need to know the current value, John
Hi John, Many thanks for your quick reply. Much appreciated. However, I am not able to understand following lines from your mail. "Now, let me tell you that X1 = 2. Now, what is the probability that X2 = 2? 0, because it is impossible that Z1 = tails. So, X2 is NOT independent of X1. ie Zn does NOT have independent increments." Your explanation will be very helpful. Ashok
X1 = Z1 - Z0, so the only way that X1 can equal 2 is if Z1 is a head (and therefore equal to 1) and Z0 is a tail (and therefore equal to -1). X1 = Z1 - Z0 X1 = 1 - (-1) X1 = 2 So, given this information, when working out the probabilities for X2, we already know that Z1 = 1. This means X2 = Z2 - Z1 X2 = Z2 - 1 Since Z2 can only be 1 or -1, it is impossible for X2 to equal 2. So P(X2 = 2 | X1 = 2) = 0 Therefore this random variable does not have independent increments.
