# True/False : Stationary process In stochastic process

• hojoon yang
In summary, the conversation discusses whether Vn = Xn / Yn is wide-sense stationary, given that Xn and Yn are independent stationary processes. The first statement is true, as E[Vn] can be simplified using the theorem E[g(x)*f(y)] = E[g(x)]*E[f(y)]. However, it is uncertain if E[1/Yn] is independent of n. For the second statement, it is stated that for Wn = Xn / Yn to be wide-sense stationary, E[Yn] must be independent of n and autocorrelations must also be independent of n. This leaves the standard deviation to be any value. An example is given where Yn has values that
hojoon yang
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Stochastic process problem!

1. If Xn and Yn are independent stationary process, then Vn= Xn / Yn is wide-sense stationary. (T/F)

2. If Xn and Yn are independent wide sense stationary process, then Wn = Xn / Yn is wide sense stationary (T/F)

I solve this problem like this:

1. E[Vn]=E[Xn/Yn], since independent E[Xn]*E[1/Yn] <- using this theorem E[g(x)*f(y)]=E[g(x)]*E[f(y)]
here, I knew it E[Xn]=μx,E[Yn]=μy, clearly not depend on 'n'

But I'm not sure E[1/yn] is not depend on 'n'

For the second one: to be wide-sense stationary, all that is required is that E[Yn] is independent of n and that autocorrelations are independent of n.
That leaves us free to make the standard dev anything we want.
Consider a very simple case where Yn can have values $3\pm \left(1+(n \mod 2)\right)$, each with 50% probability.
The mean of Yn is 3, which is independent of n. What is the mean of 1/Yn? Does it depend on n?

For the first one, write E[1/Yn] as an integral and decide whether the integral depends on n, using the fact that Yn is fully stationary, not just wide-sense.

## 1. What is a stationary process in stochastic process?

A stationary process is a type of stochastic process where the statistical properties of the process remain constant over time. This means that the mean, variance, and autocorrelation of the process do not change with time.

## 2. How is a stationary process different from a non-stationary process?

A non-stationary process is a type of stochastic process where the statistical properties of the process change over time. This means that the mean, variance, and autocorrelation of the process are not constant over time.

## 3. What are the benefits of using stationary processes in modeling?

Using stationary processes in modeling can simplify the analysis and forecasting of data. Stationary processes have well-defined statistical properties that make it easier to make predictions and identify patterns in the data.

## 4. Can a non-stationary process be transformed into a stationary process?

Yes, a non-stationary process can be transformed into a stationary process through a process known as differencing. This involves taking the difference between consecutive observations in the data to remove any trends or seasonality.

## 5. Are all stochastic processes stationary?

No, not all stochastic processes are stationary. There are different types of stochastic processes, such as non-stationary, stationary, and ergodic processes, each with their own set of characteristics and statistical properties.

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