# Expected value of random walk with optional stopping

I want to compute the expected value of the following game. I have a symmetric 1-D random walk, I can choose to stop at any time and my winnings will be the value of the random walk. What is the value of this game for large number of steps N?

I have run some simulations and have heuristic arguments that both show that it should be proportional to sqrt(N), but I'm particularly interested in the coefficient.

• what is "the value of the random walk"? The value on the number line at which you stop? Apr 2 '19 at 17:18
• Whatever X_i is equal to at step i, where X_0 = 0 and each X_i - X_(i-1) is plus or minus one with equal probability. Apr 2 '19 at 17:37

Your earnings are equivalent to $$S_N=X_1+\cdots+X_N$$, where each $$X_i$$ is iid, satisfies $$P(X_i=\pm 1)=1/2$$, has mean $$\mu=0$$ and standard deviation $$\sigma=1$$. By the CLT, $$\frac{S_N}{\sigma\sqrt{N}}=\frac{S_N}{\sqrt{N}}$$ is approximately normally distributed. So the coefficient will fluctuate roughly between -3 and 3. as that encapsulates 99.7% of all cases.
If you're interested in the absolute deviations from 0, then this is a half-normal distribution, with which gives a mean coefficient of $$\sqrt{\frac{2}{\pi}}.$$