Vol. 22, No. 1, 1967

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On the weak law of large numbers and the generalized elementary renewal theorem

Walter Laws Smith

Vol. 22 (1967), No. 1, 171–188
Abstract

{Xn} is a sequence of independent, nonnegative, random variables and Gn(x) = P{X1 + + Xn x}.{an} is a sequence of nonnegative constants such that, for some α > 0,γ > 0, and function of slow growth L(x),

∑N      αN γL(N)
ar ∼ Γ-(1-+-γ) , as N → ∞.
1

A Generalized Elementary Renewal Theorem (GERT) gives conditions such that, for some μ > 0,

       ∑            αL (x)  x γ
Ψ (x ) =   arGr(x) ∼ Γ-(1-+-γ)(μ) , as x → ∞.
(*)

The Weak Low of Large Numbers (WLLN) states that (X1 + + Xn)∕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) 1nP{Xj > n𝜖}→ 0 as n →∞, for every small 𝜖 > 0; (ii) for some 𝜖 > 0,n1 1n 0n𝜖P{Xj > x}dx is a bounded function of n. Theorem 2 supposes the {Xn} to have finite expectations and proves (*) implies WLLN if it is given that

      ℰX1--+ℰX2-+-⋅⋅⋅+-ℰXn-
limn→su∞p          n          ≦ μ,

in which case (X1 + + Xn)∕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,

1-∫ n𝜖∑n
n  0     P{Xj > x} dx → μ, as n → ∞.
1

Mathematical Subject Classification
Primary: 60.70
Milestones
Received: 11 November 1966
Revised: 30 January 1967
Published: 1 July 1967
Authors
Walter Laws Smith