A group of people share a single roll of paper towels, and when the roll runs out, the person who used the last bit is always the one to replace the roll with a fresh one from the box. The usage patterns vary from person to person; just as an example, one person might use 5 paper towels at once to clean up a big spill, but randomly and infrequently, while another person might use 1/2 of a paper towel every day as a napkin. A third person might gradually increase or decrease their paper towel usage over time. A fourth person might mimic what the third person does, always using the same amount of paper towels as the third person, immediately after the third person does so. These are just examples of the diversity of paper towel strategies. The only assumption on usage patterns is that each person uses paper towels independently of how many are left in the current roll.
Under these conditions, it seems intuitive that the long-run proportion of times a given person replaced the paper towel roll is the same as the long-run proportion of paper towels that person used.
Is this a theorem?