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| − | === Answer 3 === | + | === Answer 3 (in_progress) === |
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| + | A stochastic process X = {X_0, X_1, ..} has the stationary increment property (SIP) when, for all <math>t > s > 0</math>, the distribution of <math>X_t - X_s =</math> the distribution of <math>X_{t-s}</math>. Eg if we let t = 5 and s = 2, the SIP would ensure that distribution of <math>X_5 - X_2 =</math> the distribution of <math>X_3</math>. | ||
| + | |||
| + | Now consider the i.i.d. (independent and indentically distributed) random process <math>Z = {X_0, X_1, ..}</math>. Eg, the consecutive rolling of a fair, six-sided die would be such and i.i.d. random process. Next, we define a discrete-time sum process S as: | ||
| + | |||
| + | <math>S_n \triangleq \sum_{i=1}^{n} Z_i</math> | ||
| + | |||
| + | Therefore, our sum process X has the SIP iff sum_{t-s+1}^{t} Z_i = sum{i=1}^{t-s} Z_i for all t > s > 0. | ||
| + | Eg for t = 5 and s = 2: does the distribution of Z_5 + Z_4 + Z_3 = the distribution of Z_3 + Z_2 + Z_1? | ||
| + | |||
| + | the logic that allowed me to make the stationary increment claim leads me to make an even stronger statement: *any* (t-s) distinct i.i.d. r.v.s have an identical distribution, regardless of where they lie in the process. | ||
| + | |||
| + | We conclude that although these two sums are not generally independent (eg for the above example, a Z_3 term exists in both sums, therefore precluding independece), the two sums do possess identical distributions, therefore satisfying the SIP. this is because the process is i.i.d., therefore the distribution of *any* set of (t-s) r.v.s is equivalent. note that this last statement is a stronger statement than the SIP because it lifts the restriction that the summed r.v.s (here Z_i's) must have consecutive indices. this implies, for example, that the distribution of X_5 + X_3 + X_1 = the distribution of X_4 + X_2 + X_1. | ||
| + | |||
| + | what matters for SIP: | ||
| + | NUMBER of i.i.d. r.v.s must be same | ||
| + | INDICES of the i.i.d. r.v.s may overlap, but must be distinct within a sum | ||
| + | |||
| + | what matters for IIP: | ||
| + | NUMBER of i.i.d. r.v.s does not matter | ||
| + | INDICES of the i.i.d. r.v.s must not overlap, but may be redundant within a sum | ||
| + | |||
| + | must have same measure (in the discrete time case, the same number of time steps), and can overlap | ||
| + | |||
| + | let's illustrate this SIP for a sum process composed of i.i.d. rvs Z that represent the outcome of a fair 6-sided dice roll. | ||
| + | |||
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Revision as of 11:50, 15 April 2013
Contents
Practice Problem: Stationary Increment Property
Question
Explain what is the "stationary increment property" and why sum processes have this property.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
The stationary increment property says that the difference between the values of a random process at two distinct times will result in a random variable with the same distribution as the random variable contained in the random process at the time found by differencing the two distinct times mentioned earlier.
Sum processes have this property because the difference between the values of a sum process at two distinct times yields a random variable defined by summing the values of an iid process from the earlier time until the later time. In addition, the random variable contained in the sum process at the time found by differencing the earlier and later times is found by summing the values of the same iid process from the initial time until the time found by differencing the earlier and later times. Because the process being summed in both cases is iid, the distributions of both random variables are the same and the sum process can be said to have the stationary increment property.
Answer 2
The stationary increment property can be explained intuitively by stating that the probability distribution of any increment is only a function of how long the increment is, and not where it is located on the time domain.
For example, if it had this property, $ X_{n} $ would have the same distribution as $ X_{2n}-X_{n} $ because the length of both of the increments is n.
Sum processes have this property because the random variables in the process are time-independent. The distribution of a random variable at t = 1 is the same as a random variable at t = 2. Therefore, the important variable is the length of time that the process is allowed to run, because this is what will change the sum. Changing the start time but keeping the length of time the same will not change the distribution.
Answer 3 (in_progress)
A stochastic process X = {X_0, X_1, ..} has the stationary increment property (SIP) when, for all $ t > s > 0 $, the distribution of $ X_t - X_s = $ the distribution of $ X_{t-s} $. Eg if we let t = 5 and s = 2, the SIP would ensure that distribution of $ X_5 - X_2 = $ the distribution of $ X_3 $.
Now consider the i.i.d. (independent and indentically distributed) random process $ Z = {X_0, X_1, ..} $. Eg, the consecutive rolling of a fair, six-sided die would be such and i.i.d. random process. Next, we define a discrete-time sum process S as:
$ S_n \triangleq \sum_{i=1}^{n} Z_i $
Therefore, our sum process X has the SIP iff sum_{t-s+1}^{t} Z_i = sum{i=1}^{t-s} Z_i for all t > s > 0. Eg for t = 5 and s = 2: does the distribution of Z_5 + Z_4 + Z_3 = the distribution of Z_3 + Z_2 + Z_1?
the logic that allowed me to make the stationary increment claim leads me to make an even stronger statement: *any* (t-s) distinct i.i.d. r.v.s have an identical distribution, regardless of where they lie in the process.
We conclude that although these two sums are not generally independent (eg for the above example, a Z_3 term exists in both sums, therefore precluding independece), the two sums do possess identical distributions, therefore satisfying the SIP. this is because the process is i.i.d., therefore the distribution of *any* set of (t-s) r.v.s is equivalent. note that this last statement is a stronger statement than the SIP because it lifts the restriction that the summed r.v.s (here Z_i's) must have consecutive indices. this implies, for example, that the distribution of X_5 + X_3 + X_1 = the distribution of X_4 + X_2 + X_1.
what matters for SIP: NUMBER of i.i.d. r.v.s must be same INDICES of the i.i.d. r.v.s may overlap, but must be distinct within a sum
what matters for IIP: NUMBER of i.i.d. r.v.s does not matter INDICES of the i.i.d. r.v.s must not overlap, but may be redundant within a sum
must have same measure (in the discrete time case, the same number of time steps), and can overlap
let's illustrate this SIP for a sum process composed of i.i.d. rvs Z that represent the outcome of a fair 6-sided dice roll.
