{X_{n}} is a sequence of
independent, nonnegative, random variables and G_{n}(x) = P{X_{1} + ⋯ + X_{n} ≦ x}.{a_{n}}
is a sequence of nonnegative constants such that, for some α > 0,γ > 0, and function
of slow growth L(x),
A Generalized Elementary Renewal Theorem (GERT) gives conditions such that, for
some μ > 0,
 (*) 
The Weak Low of Large Numbers (WLLN) states that (X_{1} + ⋯ + X_{n})∕n → μ, as
n →∞, in probability. Theorem 1 proves that WLLN implies (*). Theorem 3 proves
that (*) implies WLLN if, additionally, it is given that (i) ∑
_{1}^{n}P{X_{j} > n𝜖}→ 0 as
n →∞, for every small 𝜖 > 0; (ii) for some 𝜖 > 0,n^{−1} ∑
_{1}^{n} ∫
_{0}^{n𝜖}P{X_{j} > x}dx is a
bounded function of n. Theorem 2 supposes the {X_{n}} to have finite expectations and
proves (*) implies WLLN if it is given that
in which case (ℰX_{1} + ⋯ + ℰX_{n})∕n necessarily tends to μ as n →∞. Finally, an
example shows that (∗) can hold while the WLLN fails to hold. Much use is made of
the fact that a necessary and sufficient condition for the WLLN is that, for all small
𝜖 > 0,
