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The following is paragraph (an example) from a book on stochastic differential equations that I'm reading and I would like some help in trying to understand what the author is saying.

A binary noise process $x(t)$ is defined on a $t\in J$ as follows:

(i) It takes the value $+1$ or $-1$ throughout successive intervals of fixed length $B$.
(ii) The values it takes in one interval is independent of the values taken in any other interval
(iii) All member functions differing by a shift along the $t$-axis are equally likely.

These are the assertions made by the book:

  1. A possible representation of $x(t)$ is $x(t) = y(t-A)$ with $y(t) = y_n$ on $ (n-1)B \lt t \lt nB$, where the random variable $A$ has a uniform distribution in the interval $[0,B]$.
  2. The random varibles $y_n,~n \in \{\ldots -3,-2,-1,0, 1,2,3,\ldots\}$are independent and identically distributed.
  3. The density function is given by $f(y) = \frac{1}{2} \delta(y+1) + \frac{1}{2}\delta(y-1)$.
Bobby
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  • Basically, uniformly pick a time $A$ in $[0,B]$ to "start" the process, then at each interval $B$ further forward or backward in time, randomly choose whether $x$ will be $-1$ or $1$ on the next interval of length $B$, doing so equal probability. – Ian Mar 10 '15 at 17:08
  • @Ian how do I show the assertions made by the author? – Bobby Mar 10 '15 at 20:28

1 Answers1

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The assertions are actually pretty much just restatements of the assumptions. Start by assuming that $y$ satisfies the first two assumptions and is constant on $[0,B)$. Then $y$ is constant on each interval $[nB,(n+1)B)$ for $n \in \mathbb{Z}$. Call these constants $y_n$. The second assumption implies that $y_n$ are iid, which gives the second assertion. The first assumption implies that the $y_n$ are equally likely to be $-1$ or $+1$ which is the third assertion.

To get the first assertion, note that any function which satisfies the first two assumptions is a shift of $y$ to the right by $t \in [0,B)$. But all such shifts are equally likely, which really means that the shift, which is the random variable $A$ in their notation, is uniform on $[0,B)$.

Ian
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  • Thanks. How about the third assertion? Can I say that $f(y) = \sum P(y=y_n) \delta(y-y_n)$ for all $n \in \mathbb{Z},~ y_n={-1,1}$? – Bobby Mar 11 '15 at 13:36
  • @Bobby That's just what the Dirac delta means: $\sum_{i=1}^n p_i \delta_{x_i}(x)$ is the density of the discrete variable which takes on the value $x_i$ with probability $p_i$. – Ian Mar 11 '15 at 18:17