ABSTRACT
If {Xi} is a sequence of independent random variables one would expect under suitable assumptions that for a > E[X],
P
[ X1 + · · · + XN
N ≥ a
] = 0.
Perhaps there is exponential decay and
P
[ X1 + · · · + XN
N ≥ a
] = e−NI(a)+o(N)
for some function I(a) that can be determined. We can get a bound by estimating for λ > 0,
P
[ X1 + · · · + XN
N ≥ a
] ≤ e−NλaE
] = e−Nλa
[ E [ eλX1
]]N = exp
[ −N
[ λ a − log E
[ eλX1
]]] .
