# Random jump process between {-1,1} with time delay [closed]

I want to calculate the autocorrelation function $R_{xx}$ of the signal $X\in\{-1,1\}$ that jumps with probability p between the two values:

\begin{equation*}X_{k+1} = \begin{cases} X_k &1-p \\ -X_k & p \end{cases}\end{equation*}

In addition the time step $\Delta t_{k+1} = t_{k+1} -t_k$ varies according to

\begin{equation*}\Delta t_{k} = \begin{cases} \Delta t &\mbox{if }\ X_k = 1 \\ \alpha \Delta t &\mbox{if }\ X_k = -1 \end{cases}\end{equation*}

where $\alpha \in \mathbf{N}$. Do you know how to calculate the autocorrelation function of such a process?

## closed as unclear what you're asking by Dilip Sarwate, jojek♦, Matt L., Peter K.♦Jun 22 '14 at 10:30

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Your definition of the process needs clarification. The process is a continuous-time process whose value at any time $t$ depends on its past history and which bit is being worked on right now. But you are defining it as a discrete-time process with $k$ denoting an integer in which case the time step $\Delta t$ is irrelevant. Do you want the autocorrelation of the discrete-time process or the continuous-time process? – Dilip Sarwate Jun 10 '14 at 13:54
• I changed alpha to be an integer number, I guess that was the problem. This means that the signal stays longer in the -1 state. In the limiting case $\alpha \rightarrow \infty$ at least the average should change to $-1$ as the signal would almost always display a $-1$. Does this clarify what you meant? – physicsGuy Jun 10 '14 at 14:16
• You could also reformulate my definition of the process so that if a jump from $1$ to $-1$ occurs or the signal stays in $-1$ the next $\alpha$ steps will also be $-1$. After those $\alpha$ steps a jump can again happen with probability $p$. – physicsGuy Jun 10 '14 at 14:19
• A random process is a collection of random variables $\{X_t \colon t \in \mathbb T\}$ where the index set $\mathbb T$ is usually the set of integers (or the set of nonnegative integers) or the real line (or the positive real line), etc. You need to tell us what $\mathbb T$ is. You specify $X_k$ where $k$ is clearly meant to be an integer, but you talk of switching times which may not be integer-valued, and so the process may change value from $+1$ to $-1$ at $\Delta t$ or it might stay the same. In one case, the next possible transition comes at $\Delta t + \alpha\Delta t$ while .... – Dilip Sarwate Jun 10 '14 at 14:29
• ... in the other case, the next possible transition comes at $2\Delta t$. So, it seems that the random variables $X_t$ of the process need to be defined not just at integer values of $t$ as you have done but also for other time instants, and indeed the value at any time $t$ will depend on the past history of the process. So, pick your favorite values of $\alpha$ and $\Delta t$ and tell us what the value of the process is at time $t=3.1416$. I am voting to (temporarily) close this question because it is unanswerable as it stands. – Dilip Sarwate Jun 10 '14 at 14:33