Nt jl 1 so we conclude that tnt jl and thus ntt jl t

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N(t) --+ JL. 1 so we conclude that t/N(t) JL and thus N(t)/t--+ J.l-t. Thus the long run rate of renewals is JL -t. 0
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224 7. Laws of Large Numbers and Sums of Independent Random Variables Theorem. The Glivenko-Cantelli theorem says that the em- pirical distribution function is a uniform approximation for the true distribution function . Let {Xn. n ::: 1} be iid random variables with common distribution F. We imagine F is unknown and on the basis of a sample X1, ... , Xn we seek to esti- mate F. The estimator will be the empirical distribution function (edt) defined by A 1 n Fn(X, w) =- L 1[Xj.9](W). n i=I By the SLLN we get that for each fixed x, Fn (x) F (x) a.s. as n oo . In fact the convergence is uniform in x. Theorem 7.5 .2 Theorem) Define Dn :=sup IFn(X)- F(x)l. X Then Dn Oa.s. asn oo. Proof. Define Xv,k := F+-(v/ k), v = 1, ,.,, k, where F+-(x) = inf{u: F(u)::: x}. Recall F+-(u)::: t iffu::: F(t) and (7.15) (7.16) since for any f > 0 F(F+-(u)- f) < u. If xv,k ::: x < Xv+I, k. then monotonicity implies and for such x Fn(Xv, k)- F(Xv+l , k-)::: Fn(X)- F(x) ::: Fn(Xv+I,k-)- F(Xv,k), (7.17) Since
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7.5 The Strong Law of Large Numbers for liD Sequences 225 we ·modify (7 .17) to get A 1 A Fn(Xvk)- F(xvk)- k :::;: Fn(X)- F(x) A 1 :::;: Fn(Xv+I,k-)- F(Xv+I,k-) + k' (7.18) Therefore sup IFn(x)- F(x)l X (Xvk ,Xv+l.k) A A 1 :::;: (IFn(Xv,k) - F(Xv,k)l V IFn(Xv+l.k-)- F(Xv+I, k-)1) + k' which is valid for v = 1, .. . , k- 1, and taking the supremum over v gives sup IFn(x)- F(x)l xe[xJk,xu) 1 k A A :::: k + V IFn(Xv,k)- F(xv,k)l V IFn(Xv,k-)- F(xv,k-)1 v=I =RHS . We now show that this inequality also holds for x < xu and x ::::: xu. If x :::::xu, then F(x) = Fn(x) = 1 so Fn(x)- F(x) = 0 and RHS is still an upper bound. If x <xu, either or (i) F(x) ::::: Fn(x) in which case IFn(X, w)- F(x)l = F(x)- Fn(X, w) :::: F(x):::: F(xu-) 1 <- -k so RHS is still the correct upper bound, (ii) Fn (x) > F (x) in which case IFn(X, w)- F(x)l = Fn(X, w)- F(x) :::: Fn(xu-, w)- F(xi,k-) + F(xu-)- F(x) :::: IFn(xu-, w)- F(xi , k-)1 + IF(xlk-)- F(x)l and since the last term is bounded by 1 I k we have the bound 1 A :::: k + IFn(X}k-, w)- F(xlk-)1 :::;:RHS.
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226 7. Laws of Large Numbers and Sums of Independent Random Variables We therefore conclude that Dn ::: RHS. The SLLN implies that there exist sets Av ,k. and Av ,k. such that P(Av,k) = P(Av,k) = 1, and such that and A 1 n Fn(Xv, k-) =;; L 1[Xj<Xv. k] P[Xt < Xv,k] = F(Xv,k-) 1 provided we Avk and Avk respectively. Let Ak = n Av,k n n Av ,k. v v so P(Ak) = 1. Then for w e Ak limsupDn(w)::: n->oo k lim Dn(w) = 0, n->00 7.6 The Kolmogorov Three Series Theorem 0 The Kolmogorov three series theorem provides necessary and sufficient condi- tions for a series of independent random variables to converge. The result is espe- cially useful when the Kolmogorov convergence criterion may not be applicable, for example, when existence of variances is not guaranteed. Theorem 7.6.1 Let {Xn, n ::: 1} be an independent sequence of random vari- ables. In order for Ln X n to converge a. s., it is necessary and sufficient that there exist c > 0 such that (i) Ln P[IXnl > c] < 00. (ii) Ln Var(Xn1[1Xnl=:cj) < 00. (iii) Ln E(Xn1[1Xnl=:cj) converges.
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7.6 The Kolmogorov Three Series Theorem 227 If Ln X n converges a.s., then (i), (ii), (iii) hold for any c > 0. Thus if the three series converge for one value of c > 0, they converge for all c > 0.
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