The Asymptotic behavior of the Final Location of a One-Dimensional (LUCKY!) \
Walker with Step Set, {-2, -1, 1, 2},
That Never Goes to the Left of 0 After n steps
as n gets larger and larger
By Shalosh B. Ekhad
Theorem: Consider a walker on the discrete line, that starts right next to a\
n abyss, such that if he ever goes to the left
of the starting point, he would fall and die.
Each minute, he
picks with EQUAL probability, i.e. each with prob., 1/4
one of the steps in the set, {-2, -1, 1, 2}
of course with very high probability, sooner or later he would fall off the \
cliff and die.
Suppose, however, that the walker is a lucky one, and is still alive after n\
minutes.
Conditioned on this happy fact, consider the random variable: location after\
n minutes.
Equivalently, consider a gambler in a FAIR casino, where at each minute, wit\
h probability, 1/4, she gets one of the amounts in the set,
{-2, -1, 1, 2}
Since the amounts add up to 0, it is a fair casino. She starts with 0 dollar\
s, and would get kicked out as soon as she owes the casino money.
Assuming that she is still at the casino after n minutes, let X(n) be the ra\
ndom variable, amount owned by the gambler after n minutes.
The expectation of X(n) is asymptotically ,
1/2 1/2 1/2
n 5 Pi 0.66496166 0.0357769
--------------- - 1.2763932022 + ---------- - ---------
2 1/2 n
n
and in Maple input format
1/2*n^(1/2)*5^(1/2)*Pi^(1/2)-1.2763932022+.66496166/n^(1/2)-.357769e-1/n
The variance of X(n) is asymptotically ,
16.344761 109599.
1.073009181641 n - 0.32947657943 + --------- - -------
n 2
n
and in Maple input format
1.073009181641*n-.32947657943+16.344761/n-109599./n^2
Note that we do NOT have concentration of mass.
The skewness (aka standardized third moment (about the mean)) is asymptotica\
0.90109238 0.267608
lly , 0.63111065780 - ---------- - --------
n 2
n
and in Maple input format
.63111065780-.90109238/n-.267608/n^2
The kurtosis (aka standardized fourth moment (about the mean)) is asymptotic\
3.3407915 22.1939
ally , 3.2450892988 - --------- + -------
n 2
n
and in Maple input format
3.2450892988-3.3407915/n+22.1939/n^2
This ends this article that took, 294.920,
to generate. I hope that you enjoyed it.