I was thinking about the discrete random variable describing the stopping time ($T$: random variable modelling the toss number where he first reaches his target) of a wealthy gambler reaching his target. It is discussed in some detail here: Gambler with infinite bankroll reaching his target and here: Probability that random walk will reach state $k$ for the first time on step $n$. I realized that when the coin is biased against the wealthy gambler, there is a finite chance he will never reach his target. So, if you calculate the summation:
$$\sum_{t=0}^\infty P(T=t)$$
you will only get $1$ if the coin he is using has a probability, $p\geq \frac 1 2$ of heads. Otherwise, the summation above will result in a number less than $1$. Looking at the definition on Wikipedia, no where does it say that the probability mass function should sum to $1$ (emphasis: in the formal definition). However, right outside the scope of the formal definition, it does.
But this would imply that the wealthy gamblers stopping time when $p < \frac 1 2$ has no PMF?
Just wanted to get the community's opinion on this.
Also, if we conclude the PMF doesn't have to sum to $1$, is there then any example of a corresponding probability density function that doesn't integrate to $1$? Perhaps the stopping time (defined as reaching a positive boundary) of a continuous time random walk with negative drift?
EDIT: saying that "never reaching the target is included in the possible outcomes" is not satisfying. We are talking about the random variable $T$. This random variable has a certain domain (which includes $\infty$). Summing over the domain should give you $1$. Where in its domain should we fit "never reaching the target"? The fundamental problem remains, is $P(T=t)$ the PMF of $T$ or not? If we say it isn't because it doesn't sum to $1$ over all possible values of $T$, then does it mean $T$ doesn't have a PMF?
Let's say that $s=\displaystyle \sum_{t=0}^\infty P(T=t)\ne 1$ (which is, as you say, the case when $p<\frac12$ and it is possible that the target will never be reached.) In this case when we talk about the probability that $T=t$, (in other words, the probability that the target is reached in $t$ games), we are implicitly assuming that the target is actually reached. Since the probability of that is $s$, the probability that the target is reached in $t$ games, given that the target is reached, is $\frac 1 s P(T=t)$, which, of course, sums to 1 as $t$ runs through its domain.
In my comment, I mentioned that saying an arbitrarily large integer is in the domain of a function is not the same as saying that $\infty$ is in the domain. Notice that $\infty$ is not in the domain of the $P$ in your post. If we define $t=\infty$ to mean that the target is never reached, and then define the domain of a mass function $P^\prime$ to be $\mathbb Z^{\ge0}\bigcup \left\{\infty\right\}$, and $P^\prime$ to be:
$$P^\prime(T=t)=\left\{\begin{array}{rl}P(T=t),&t\in\mathbb Z^{\ge0}\\ 1-s,&t=\infty\end{array}\right.$$
where $P(T=t)$ is the function in your post and $s$ is as I've defined it above, then this mass function $P^\prime$ does sum to 1.
You might find some of my struggles with the probabilities not summing to 1 in a related problem in this thread interesting.