# Can you prove a random walk might never hit zero without the countable additivity axiom?

Can you prove a random walk might never hit 0, given a probability system that only uses the finite additivity axiom (rather than the standard countable additivity axiom)?

Specifically: Imagine the standard random walk problem on the nonnegative integers: We start at integer location $i>0$. Every step we independently move left with probability $\theta$, and right with probability $1-\theta$. Assume $0<\theta < 1/2$. Let $q$ be the probability that we ever hit 0, given we start at location 1. By the repeated structure of the problem, we can infer:
$$q=\theta + q^2(1-\theta)$$
The difficulty is that this quadratic has two roots: $q=1$ and $q=\frac{\theta}{1-\theta}$. If we can prove that $q<1$, then we infer $q=\frac{\theta}{1-\theta}$.

The answer is $q=\theta/(1-\theta)$ under the standard probability axioms, which includes the countable additivity axiom. Is this provable with only finite additivity, i.e., $P[\cup_{i=1}^k A_i] = \sum_{i=1}^k P[A_i]$ for finite integers $k$ and for disjoint events $A_i$? Or is it undecidable?

Note: The union bound $P[\cup_{i=1}^{\infty} A_i] \leq \sum_{i=1}^{\infty} P[A_i]$ is unprovable without countable additivity.

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How would you define the finitely additive probability law of the random walk?

With a standard probability model, the random walk could, for instance, be defined in a probability space $(\Omega,\mathscr{F},\mathbb{P})$, where $\Omega:=\{-1,1\}^{\mathbb{N}}$ with $x\in\Omega$ specifying the sequence of moves, $\mathscr{F}$ is the product $\sigma$-algebra and $\mathbb{P}$ is the joint probability law of the random moves (a product measure). The intuitive concept of a random walk only prescribes the finite-dimensional marginals of $\mathbb{P}$ (i.e., the measure of the cylinder sets) and one would rely on Carathéodory’s extension theorem (or something similar) to be sure that the probability of every event in $\mathscr{F}$ is determined uniquely and consistently.

If you discard the countable additivity axiom, then it is not clear that the probability of the event $E:=\{\text{the RW eventually hits the origin}\}$ is uniquely determined by the finite-dimensional marginals. We know there is at least one model (the countably additive one) in which $\mathbb{P}(E)=q$, but there might be other non-countably additive models in which $\mathbb{P}(E)$ has a different value.