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Chapter 2 P20

R

rsmallela

Member
3.2 General Random Walk.

What is Stationary independent increment??

Independent increment is explained in defn on p-15.

Thanks,
Raj
 
rsmallela said:
3.2 General Random Walk.

What is Stationary independent increment??

Independent increment is explained in defn on p-15.

Thanks,
Raj
Click to expand...


Hi Raj, it's been over a year and half since I sat CT4 but here goes: :)

The independent property of a random walk just means that any two non-overlapping time periods are independent!

So for example, X_t to X_t-1 is independent of any t greater than or equal to one. (This is just maths speak. So X_4 to X_5 is independent of X_5 to X_6 etc)

The stationary property of a random walk just means that it does not depend on time, i.e. it is constant. (Formally, I vaguely remember the notes saying that the mean is constant and the covariance depends on the lag) So,

If we are told that the increment of a random walk is normally distributed with mean mu and variance sigma squared then this distribution is the same for ALL increments (Hope that helps).

Does anyone disagree?

Good luck!!
 
Last edited:
Thanks

So, when we see 'Independent Stationary' Increment - it means, the increment is independent of time and the statistical properties remain same all the time. Am I right now?

Thanks,
Raj
 
Chapter 3 P20

Page 20: Model 5.3 Simple Random Walk on S...

Can some one explain the three equations under 'The Markov property holds:'...??

Thanks for your help.

Thanks,
Raj
 
If a process is stationary (we usually mean weakly) then the mean is constant and the covariance depends ONLY on the lag.

If a process has stationary increments then it means that the mean of the increments is constant and the covariance of the increments depends ONLY on the lag.

If a process has independent increments then the statistical distribution of those increments Xt - Xs is completley independent of the history of the process up until time s.

If a process has stationary, independent increments then it does both the last 2 things.
 
Raj,

Try and post your questions as separate posts so we know which one is being answered. Your one about page 20...

Those 3 equations show that the probability distribution for future states is the same whether we know just the current value or the entire history of the process - this is Markov by defiinition,

John
 
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