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So I think a logarithmic average will do the trick for you. If you define $Y_n$ to be the sign of $a_1+\ldots+a_n$, then calculations with Brownian motion in place of random walk suggest the covariance of $Y_n$ and $Y_m$ with $m<n$ is approximately $(1/2\pi)\arctan\sqrt{m/m-n}$. Now define $S_N=(1/\log N)(Y_1/1+\ldots +Y_N/N)$. This has expectation 0 and variance $\approx 1/\log N$, which gives a systematic way of saying that the random walk is "positive half the time".