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It is given in Papoulis that:

Two random processes X(t) and Y(t) are equal in the MS sense iff \begin{equation} E{|\mathbf X(t)-\mathbf Y(t)|^2} = 0 \end{equation} for every t.

It follows that \begin{equation} \mathbf X(t, \xi) = \mathbf Y(t,\xi) \end{equation} with probability 1 where $\xi$ is an outcome.

I have two questions here:

1) What do you mean by two random processes are equal with probability 1 and how is it different from two random processes are equal i.e, X(t,$\xi$) = Y(t,$\xi$), $\forall\xi$?

2) How does the first equation implies second?

The exact text given in the book:

Two processes X(t) and Y(t) are equal in MS sense iff $$ E{|\mathbf X(t)-\mathbf Y(t)|^2} = 0 $$ for every t. Equality in the MS sense leads to the following conclusion: We denote by $A_t$ the set of outcomes $\xi$ such that $\mathbf X(t,\xi)=\mathbf Y(t,\xi)$ for a specific t, and by $A_\infty$ the set of outcomes $\xi$ such that $\mathbf X(t,\xi)=\mathbf Y(t,\xi)$ for every t. From the above equation it follows that $\mathbf X(t,\xi)-\mathbf Y(t,\xi) = 0$ with probability 1; hence $P(A_t)=P(S)=1$. It does not follow, however, that $P(A_\infty)=1$. In fact, since $A_\infty$ is the intersection of all sets $A_t$ as t ranges over the entire axis, $P(A_\infty)$ might even equal 0.

I'm facing difficulty in understanding/visualizing the above. Thank you very much for the reply for the previous questions which helped me understanding half way. Can you give me an example for the following which I hope will help me understand the above conclusion.

Let two sets $A_{t1}$ and $A_{t2}$ are defined as: $$A_{t1} = \{\xi:\mathbf X(t1,\xi)=\mathbf Y(t1,\xi)\}$$ $$A_{t2} = \{\xi:\mathbf X(t2,\xi)=\mathbf Y(t2,\xi)\}$$

MS equality implies: $$P(A_{t1}) = 1$$ $$P(A_{t2}) = 1$$

Please give an example of $\mathbf X(t,\xi)$ and $\mathbf Y(t,\xi)$ such that $P(A_{t1}\cap A_{t2})=0$

2 Answers 2