What is the mean of "limit goes to zero uniformly in $s$ as $t$ goes to infinity"?

Question

This is a very simple question, but I do not understand the concept. Consider $P(s,t)$ and $L$ as two stochastic matrices, where $s$,$t$ are time variables and $P(s,t)$ is the probability of the occurrence of an event at time $t$ with considering an event at time $s$. Also define norm of $P(s,t)-L$ as follows: $$||P(s,t)-L||=\sup_i \sum_j^\infty|a_{ij}|$$ What is the mean of: "$||P(s,t)-L||$ goes to zero uniformly in $s$ as $t$ goes to infinity" ?

Answer 1

It means that as $t\to \infty$, the expression $\|P(s, t)-L\|$ goes towards $0$, and it does so uniformly in $s$. This means in a sense that as $t$ increases, the expression goes to $0$ for all different $s$ simultaneously.

I think this is easiest illustrated with a counterexample (using $1\times 1$ matrices, also known as real numbers). Consider $f(s, t) = e^{-(s+ t)^2}$. For any fixed $s$, as $t\to \infty$, the value of $f(s, t)$ goes toward $0$. However, for any fixed $t$ there is always somewhere where $f$ is far from $0$ (its max value is $1$). No matter how large $t$ gets, we can't get rid of that max value of $1$. We can only push it further and further out to the side. So $f(s, t)$ goes to $0$ as $t$ goes to $\infty$, but it doesn't do so uniformly in $s$.

"Converging uniformly in $s$ as $t$ goes to $\infty$" can therefore be intuitively explained as "not only does increasing $t$ make any single point on the function go towards $0$, but it makes the function as a whole (especially including max and min points) go toward $0$."

There is, of course, a formal "$\epsilon$-$\delta$" definition if you'd like:

We say that $f(s, t)$ converges to $g(s)$ uniformly in $s$ as $t\to \infty$ iff for any $\epsilon<0$ there is an $N\in \Bbb R$ such that for any $t > N$ and for any $s\in \Bbb R$ we have $$ \|f(s, t)-g(s)\| <\epsilon $$