I recall a documentary in which a slot machine had trial runs and at first, the desired "bingo" outcome came out more often, but later waned into losses. A scientist plotted the graph, a discrete function with a maximum and then its value decreases to zero as n approaches infinity.
I incidentally am studying second order linear differential equations and I think this phenomenon could be a case of such DEs. I wonder how correct my intuition is.
I would think that you need some psychology and two functions here.
The first function would describe beginners luck, and that would be that for some 'win' function $w(t)$ defined as the (expected) maximum profit at any time before $t$. If you'd take $\frac{w(t)}{t}$, you'd see large values for small $t$, that's why it is called beginners luck.
The second function would be loss $l(t)$, the loss at time $t$ (expected, maximum, or current loss).
The $w(t)$ function is a positive function, and the $l(t)$ function is a non-negative function. For continuous time/probability you could describe $w(t)$ as D.E.
And of course, psychology of gamblers considers $w(t)$ as the more likely outcome, though $l(t)$ is.
Gamblers act on $w(t)$, thinking that $lim_{t->\infty}w(t) = \infty$ so infinite profit if you play long enough. And maybe bad gamblers even think that if they play well, they can make a profit at a rate of $max(\frac{w(t)}{t})$.