Question
Using this definition of a random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
I have an intuitive notion of a property of a random sequence. That the sequence should remain random after "random" series of operations (depending on the entries of the sequence) of addition and modulus subtraction (modulus). What is this notion/property of randomness called? Does it hold?
Example
Let the $n$'th term be $a_n$ then $b_n = |a_n - a_{n+1}| - a_{n+1} $
If $a_n$ is a random sequence, so is $b_n$
It is not precisely clear what definition of randomness you use when you say a sequence of numbers "is random" or "stays random". But I assume you mean something like the following:
Let $n \in \mathbb{N}_{\geq 2}$ and let $X_1,X_2,... \Omega : \rightarrow \{1,...,n\}$ be independent, discrete random variables and let each $X_i$ be uniformly distributed on the set $\{1,...,n\}$. Then, for each $w \in \Omega$, the sequence $$ s(\omega):= (X_1(w),X_2(\omega),...) $$ could be called a random sequence. This is the most straightforward definition of randomness.
If you would now (say) use addition of components to form a new sequence $$ s_{+}(\omega) := (X_1(\omega)+X_2(\omega), X_2(\omega)+X_3(\omega),...) ,$$ then all entries of $s_{+}$ are still identically distributed, but they have lost their independence. To see this, imagine that $X_i(\omega)+X_{i+1}(\omega)$ is very high. This would imply an increased probability that $X_{i+1}(\omega)$ is very high, which in turn would imply an increased probability that the next entry $X_{i+1}(\omega)+X_{i+2}(\omega)$ is very high as well.
Since the sequence $s_+$ has no independent entries anymore, it is in this sense less random then the sequence $s$. As a rule of thumb: the more operations (addition, subtraction, division of components,...) you apply to $s$, the more dependent the entries of the resulting sequence will get and the more patterns will emerge, which will make the sequence less random in an intuitive sense.
The point is, if you accept the definition from above about what it means for a sequence $s$ of natural numbers to be random, then it is impossibe for the sequence $s_+$ to be random as well according to the same definition of randomness. The same holds for similar sequences $s_-,s_*,...$. As soon as the entries of the sequence loose their independence, patterns can and will emerge in the long run.