Suppose a stochastic process $X = \{X_t, \, 0 \leq t \leq n \}$.
I have seen some of my friends writing the random variable $X_t$ as:
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~X_t = X_t \, \sum_{i \, = 0}^{n} \mathbb{1}_{\{t \, = \, i\}} = \sum_{i \, = \, 0}^{n} X_i \, \mathbb{1}_{\{t \, = \, i\}}\,$,
where $\mathbb{1}_{\{t \, = \, i\}}$ is the indicator function that gives $1$ if the event $\{t \, = \, i\}$ is true, and $0$ otherwise.
Is the above expression correct? If correct, why? can someone explain to me in detail what is happening? why for example the sum is used and not another operation?
Any help will be very appreciated!
Since neither $\ i\ $ nor $\ t\ $ is a random variable, $\ \{t=i\}\ $ is not an event in your probability space, and $\ \mathbb{1}_{\{t=i\}}\ $ isn't a random variable. The only sensible meaning I can think of for $\ \{t=i\}\ $ is a singleton set containing the truth-value "true" if $\ t=i\ $, or the truth-value "false" if $\ t\ne i\ $, and $\ \mathbb{1}_{\{t=i\}}\ $ would then simply be Kronecker's delta. In that case, the equations $$ X_t=X_t\sum_{i=0}^n \mathbb{1}_{\{t=i\}}=\sum_{i=0}^n X_i\mathbb{1}_{\{t=i\}} $$ are true because $\ \mathbb{1}_{\{t=i\}}=1\ $ if and only if $\ i=t\ $ and is $0$ otherwise, so the only term that contributes to either sum is the $\ t^\text{th}\ $.