Random walk and the arc-sine laws

Key words: The first arc-sine law, the second arc-sine law, coin tossing, random walks

The first arc-sine law, the second arc-sine law, coin tossing, random walksBrownian motion can be treated as a limit of random walks. Random walk can be simulated by tossing a fair coin. Suppose that there is an ideal coin tossing game in which each player wins or loses a constant amount with probability 0.5 at each throw. By the law of Large Numbers, the probability of positive outcome is approaching 0.5 as the number of games increases. The final result is likely to fluctuate changing sign from positive to negative and back. How often the path of such a random walk crosses the x-axis, i.e. what is the fraction of time spent on the “positive” side vs. “negative” side?

Taking into account the law of Large Numbers and the obvious symmetry in the time between two consecutive draws it is equally likely to win or lose. With increasing duration of the walk more draws will occur which seem to imply that the fraction of time spent on the “positive” side is 0.5.
But it may come as a complete surprise that the path crosses the x-axis rarely. Moreover, as more games are played, the frequency of crossings decreases. This is perhaps the most counterintuitive aspect of Brownian motion.

However, there are principles that pertain to random walks. These are the arc-sine laws. They assume that the amount to win is equal to the amount that can be lost and that this is always a constant amount. The arc sine laws also assume that there is a 50% chance of winning and a 50% chance of losing, i.e. a game has the mathematical expectation that is zero.

The first arc-sine law states that the probability that in N games the fraction of time spent in the winning zone tends to:

1 / (Pi *(k*(N-k)) ^ 0.5)

as k and (N-k) approaching to infinity, where k is the number of the events spending in positive side.

After applying some mathematical rules, the cumulative probability that the fraction of time spent on the positive side is less than X (with 0 < X < 1) is

2/Pi * ARC SIN (X ^ 0.5)
The arcsine term gives this rule its name, the arcsine law.

By simulation it can be seen that surprisingly long series with positive or negative results can occur. This can explain why 50% of traders consistently lose money while 50% consistently win.

The second arc-sine law states that the maximum points will most likely occur at the endpoints and least likely at the center. Using the same formula as above, the probabilities of being in positive territory for K after 10 tosses (N = 10):


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