Journal of Mathematical Sciences, Vol. 132, No. 5, 2006

A NOTE ON BLACKWELL’S RENEWAL THEOREM

E. A. M. Omey 1 and J. L. Teugels 2

UDC 519.2

1. Introduction Our point of departure is a renewal process for which we define a number of crucial quantities. All of these concepts can be found in [1, 5, 7, 10]. Definition 1. Let {Xi , i ∈ N} be a sequence of independent identically distributed (i.i.d.) random variables with common distribution F of X, where X ≥ 0 but X ≡ 0. The sequence {Xi , i ∈ N} is called a renewal process. Let S0 = 0 and Sn = Xn + Sn−1 , n ≥ 1. The sequence {Sn , n ∈ N} constitutes the set of renewal (time) points. Let also t ≥ 0, and define N (t) = sup[n: Sn ≤ t]; then the processes {N (t), t ≥ 0} and N0 (t) := 1 + N (t) are called the renewal counting processes. The renewal functions are defined and denoted by U (t) =: EN (t) and U0 (t) =: I(t) + U (t). Finally, we write µ := EX ≤ ∞. We call the processes of the associated functions processes generated by F or by X. Note that N0 (t) is a stopping time for the sequence {Xi , i ∈ N}. Definition 2. Let {Si , i ∈ N} and {N (t), t ≥ 0} be as defined above. For every t ≥ 0, we define (i) the age of the renewal process by Z(t) = t − SN (t) ; (ii) the residual life (time) of the renewal process by Y (t) = SN (t)+1 − t = SN0 (t) − t. We discuss the precise relationship between the asymptotic behavior of the age or residual life and that of the difference process D(t, y) := N (t + y) − N (t) = N0 (t + y) − N0 (t). For µ < ∞, we show that if t ↑ ∞, then Blackwell’s theorem, ED(t, y) → y/µ, is equivalent to the convergence in distribution of Y (t) or of Z(t) to a nondegenerate limit. We also give a result for the case where µ = ∞. The technical link between age and residual life time is expressed by the following equation: t−y
P{Y (t) ≥ y, Z(t) ≥ z} = (1 − F (t + z − u)) dU0 (u),

t ≥ y,

(1)

0

which, in particular, implies that P{Y (t) ≥ y, Z(t) ≥ z} = P{Y (t + z) ≥ y + z} = P{Z(t − y) ≥ y + z}.

(2)

2. The Finite Mean Case D

From (1) and the key renewal theorem, one infers that for F nonlattice and µ < ∞, Z(t) −→ Z ∗ . More explicitly, for all u ≥ 0,
u 1 ∗ lim P{Z(t) ≤ u} = P{Z ≤ u} = (1 − F (v)) dv = µ−1 m(u), (3) t↑∞ µ 0

where


x (1 − F (y)) dy

m(x) := 0

is the integrated tail of F . D D From (2) it follows that (3) holds if and only if Y (t) −→ Y ∗ (or even (Y (t), Z(t)) −→ (Y ∗ , Z ∗ )), where P{Y ∗ ≥ y, Z ∗ ≥ z} = P{Y ∗ ≥ y + z} = P{Z ∗ ≥ y + z}. For the finite mean case we have the following result. THEOREM 1. Suppose that F is nonlattice and has finite mean µ. Then the following statements are equivalent: (2.1) limt↑∞ {U (t + y) − U (t)} = y/µ; D

(2.2) Z(t) −→ Z with Z nondegenerate. Proceedings of the Seminar on Stability Problems for Stochastic Models, Pamplona, Spain, 2003, Part II. 656

c 2006 Springer Science+Business Media, Inc. 1072-3374/06/1325-0656

Either statement implies that Z and Z ∗ have the same distribution. Proof. Fix y ≥ 0 during the following proof. Together with the original renewal process we define a shadow renewal process {Xn , n ≥ 1} generated by X  with the same distribution F as the original process. The renewal quantities Sn , N  (t), N0 (t), U  (t), U0 (t), Y  (t), and Z  (t) are defined as for the first process. We first prove two auxiliary results. LEMMA 1. Processes D(t, y) and N0 (y − Z(t)) have the same distribution. Proof. We need to show that for all integral values of k ≥ 0 P{D(t, y) = k} = P{N0 (y − Z(t)) = k}. For convenience, write q(k) = P{N0 (t) = k, D(t, y) = k}. For k = 0

q(n, 0) = P{N0 (t) = n, y − Z(t) < 0}.

Further, q(n, 1) = P{N0 (t) = n, Sn ≤ t + y < Sn+1 } = P{N0 (t) = n, 0 ≤ y − Z(t) < Xn+1 }. Since the event {N0 (t) = n, Z(t) ≤ z} is independent of Xn+1 , we have q(n, 1) = P{N0 (t) = n, 0 ≤ y − Z(t) < X1 } = P{N0 (t) = n, N0 (y − Z(t)) = 1}. For k ≥ 2 we have, similarly, q(n, k) = P{N0 (t) = n, Sn+k−1 ≤ t + y < Sn+k }  = P{N0 (t) = n, Sk−1 ≤ y − Z(t) < Sk } = P{N0 (t) = n, N0 (y − Z(t)) = k}. Summing over n gives the required result. D

LEMMA 2. If F is nonlattice and µ < ∞, then D(t, y) −→ N0 (y − Z ∗ ). Proof. As in the previous proof, first take k = 0. Then, by the previous lemma, P{D(t, y) = 0} = P{N0 (y − Z(t)) = 0} = P{y − Z(t) ≤ 0} −→ P{y − Z ∗ ≤ 0} = P{N0 (y − Z ∗ ) = 0}, where we used (3). For k > 0 we show that

P{D(t, y) = k} −→ P{N0 (y − Z ∗ ) = k}.

Using (3) and the lemma again, we have for k > 0 P{D(t, y) > k} = P{N0 (y − Z(t)) > k} = P{Z(t) ≤ y − Sk } −→ P{Z ∗ ≤ y − Sk } = P{N0 (y − Z ∗ ) > k}. This proves the lemma. Proof of Theorem 1. We only need to prove that (2.2) implies (2.1), since the other implication follows from (3). Now, Lemma 1 implies the following explicit expression:
y P{Z(t) ≤ y − v} dU0 (v) =: Gt ∗ U0 (y),

ED(t, y) =

(4)

0

where we use the notation ∗ for a convolution and the abbreviation Gt (z) = P{Z(t) ≤ z}. On the right-hand side, we can take the limit for t ↑ ∞. Indeed, Gt (z) → G(z) := P{Z ≤ z} by assumption and, hence, dominated convergence applies. Therefore, lim ED(t, y) =: k(y)

t↑∞

657

exists. Obviously, 0 ≤ k(y) is nondecreasing and satisfies the functional equation k(u + v) = k(u) + k(v). Hence, k(u) = βu for some nonnegative quantity β. That β = µ−1 follows from the elementary renewal theorem and Ces`aro’s ˆ ˆ0 (s) = ˆ U ˆ0 (s) for the Laplace–Stieltjes transforms. Since U lemma. Alternatively, k(y) = G ∗ U (y) leads to k(s) = G(s) ˆ ˆ (1 − Fˆ (s))−1 with Fˆ the transform of the generating distribution β/s = G(s)(1 − Fˆ (s))−1 , G(s) = β(1 − Fˆ (s))/s. Since ˆ Z is nondegenerate, β > 0. Furthermore, since G(0) = 1, β = µ−1 . That Z and Z ∗ have the same distribution is obvious. While Blackwell’s theorem shows that lim ED(t, y) = y/µ, Lemma 2 shows that D(t, y) itself converges weakly to N0 (y − Z ∗ ). 3. The Infinite Mean Case Since the derivation of the results of [3] and [8] (see also [2]), it is traditional to assume that the renewal process is generated by a distribution with regularly varying tail, i.e., for some slowly varying function (x) and an index of regular variation α ∈ (0, 1) 1 − F (x) ∼ x−α (x) for x ↑ ∞. A slightly more appropriate formulation goes in terms of the integrated tail m(x). Denote the class of functions of regular variation with index β by Rβ . Let Z ∗ (α) denote a positive random variable with density pα (x) = sin(πα)(πxα (1 + x))−1 on x > 0. The following result is almost classical and of elementary renewal type. LEMMA 3. Let (t) be slowly varying and 0 < α < 1. Then as t ↑ ∞ the following conditions are equivalent: (i) m(t) ∈ R1−α ; (ii) U (t) ∈ Rα ; D (iii) Z(t)/t −→ Zα∗ . The proof can be found by a combination of results in [2, Sec. 8.6.2]. A more general class than the class of regularly varying functions is the class L of positive, measurable functions a on R+ , for which limt↑∞ a(t + y)/a(t) = 1 locally uniformly in y. We link the quantity D(t, y) to the residual life through the following result. THEOREM 2. Suppose a(t) ∈ L. Then the following statements are equivalent: (3.1) limt↑∞ a(t)(U (t + y) − U (t)) = k(y) for all y > 0; (3.2) limt↑∞ a(t)P{Z(t) ≤ y} = v(y) for all y > 0. Moreover, if (3.2) holds, then k(y) = βy for some β ≥ 0 and v(y) = βm(y). Proof. That (3.2) implies (3.1) follows from (4). Indeed,
y a(t)P{Z(t) ≤ y − v} dU0 (v),

a(t)E(D(t, y)) = 0

and (3.1) follows from Lebesgue’s theorem. The limit equals k(y) = v ∗ U0 (y). Before embarking on the converse, note that the existence of the limit in (3.1) implies its specific form. By the L-condition on a, one immediately sees that the limit k satisfies the functional equation k(u + v) = k(u) + k(v). Since, furthermore, k(u) ≥ 0, k(y) = βy for some β ≥ 0. For the converse we start from the distribution of Z(t). Take y ≤ t. It is not difficult to derive from (1) that
y P{Z(t) ≤ y} = (1 − F (y))(U0 (t) − U0 (t − y)) +

(U (t) − U (t − u)) dF (u). 0

But then (3.1) easily implies that a(t)P{Z(t) ≤ y} converges. The limit v(y) is given by
y v(y) = (1 − F (y))k(y) +


y (1 − F (u)) dk(u).

k(u) dF (u) = 0

0

Since k(u) = βu, it follows that v(y) = βm(y). Theorem 3 is proved. We make a number of remarks with respect to the last result. The L-condition is not used for the implication (3.2) to (3.1). Further, in (3.1) and (3.2), the limits k(y) and v(y) can be replaced by O(1)-expressions. For general results in such a direction, see [9]. 658

An immediate application of the above theorem results from a Blackwell-type theorem (see [4]), where the function a(x) = m(x) is regularly varying with index α ∈ (1/2, 1]. Of course, if a(x) is constant, then we recover part of Theorem 1. 4. The Lifespan of a Renewal Process It would be an interesting adaptation of the results if we could prove the following results. Let L(t) := Y (t) + Z(t) be the lifespan of a renewal process. Alternatively, L(t) = SN0 (t) − SN (t) = XN0 (t) . It is clear that, in the finite-mean D

case, the conditions of Theorem 1 imply that also L(t) −→ L∗ , where ∗

−1


y

P{L ≤ y} = µ

v dF (v). 0

This follows from the relation ⎧ ⎨ 1 − (1 − F (y))U0 (t) t P{L(t) ≤ y} = ⎩ (F (y) − F (t − x)) dU (x) t−y

if t ≤ y, if y ≤ t

and the key renewal theorem. D The reciprocal implication that L(t) −→ L∗ (not constant) implies Blackwell’s result is less obvious. We have had no success in trying to prove this. Relations that might help and that link L(t) with the quantity D(t, y) are only in integral form and go the wrong way. For example, the following facts are known: E exp{−θL(t)} = Kθ ∗ U0 (t), where


∞ Kθ (x) =

e−θv dF (v).

x

This, together with the key renewal theorem, also proves the result on L∗ . A further transform with respect to t is possible, but then it does not seem obvious how to use the fact that t ↑ ∞. For results on L(t), see [6] or [11]. REFERENCES 1. G. Alsmeyer, Erneuerungstheorie, Teubner, Stuttgart (1991). 2. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge (1987). 3. E. B. Dynkin, “Some limit theorems for sums of independent random variables with infinite mathematical expectations,” Selec. Trans. Math. Statist. Probab., 1, 171–189 (1961). 4. K. B. Erickson, “Strong renewal theorems with infinite mean,” Trans. Amer. Math. Soc., 185, 371–381 (1970). 5. W. Feller, An Introduction to Probability Theory and Its Applications, Vol.2, Wiley, New York (1971). 6. K. Hinderer, “A unifying method for some computations in renewal theory,” Z. angew. Math. Mech., 65, 199–206 (1985). 7. S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, Academic Press, New York (1975). 8. J. Lamperti, “An invariance principle in renewal theory,” Ann. Math. Statist., 33, 685–696 (1962). 9. E. Omey, “On a subclass of regularly varying functions,” J. Statist. Plan. Inf., 45, 275–290 (1995). 10. S. M. Ross, Stochastic Processes, Wiley, New York (1983). 11. J. L. Teugels, “The lifespan of a renewal,” in: Stochastic Musings, Festschrift for C. Kevork and P. Tzortopoulos, Institute of Statistical Documentation Research and Analysis, Athens, Greece (2000) (to appear). 1 2

EHSAL, Stormstraat, 1000 Brussels, Belgium. E-mail: [email protected]. University Centr for Statistics, DeCroylaan, 52B, 3001 Heverlee, Belgium. E-mail: [email protected].

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