Convergence Rates and Limit Theorems for the Dual Markov Branching Process

Abstract

This paper studies aspects of the Siegmund dual of the Markov branching process. The principal results are optimal convergence rates of its transition function and limit theorems in the case that it is not positive recurrent. Additional discussion is given about specifications of the Markov branching process and its dual. The dualising Markov branching processes need not be regular or even conservative.

1. Introduction

The Markov branching process (MBP) is the discrete-state Markov process on the state-space whose transition function is standard and satisfies the branching property, where the asterisk denotes convolution. We allow the possibility that is dishonest, . We shall see in the next section that is stochastically monotone; that is, is nondecreasing in , for each . Consequently, as we explain in Section 3, it follows from the principal result (Siegmund’s theorem) in [1] that the following definition of a dual MBP (DMBP) is not vacuous.

Definition 1. A DMBP is an -valued Markov process whose transition function satisfies This process is the focus of the present paper. Definition 1 comes from [2] which also gives two equivalent definitions, one based on the -matrix and one based on a so-called dual branching property ((22) below). We say a little about these definitions in Section 3. It is shown in [2] that is regular and hence that is minimal and honest. That paper also gives criteria for recurrence and transience. In particular the DMBP is positive recurrent if and only if the MBP is supercritical and the zero state is accessible, in which case the limiting-stationary law of the DMBP is geometric. A curious property is that is strongly ergodic if and only if is dishonest.

There are two principal contributions of the present paper. The first is a study of the convergence properties of the transition function . If the zero state is inaccessible for the MBP, then it is absorbing and accessible for the DMBP. In this case there is a limiting conditional law (Theorem 3). In the positive recurrent case, Theorem 4 gives optimal convergence rates for the variation distance between and its limiting geometric law. A by-product is an endogenous proof of the curious property mentioned above. Theorem 5 gives the exact rate at which in the case that the MBP is subcritical, and Theorems 6 and 8 deal with the critical case. These results follow from convergence rate results about , the latter being fairly well known, at least when is honest.

The second contribution is an account of limit theorems for the DMBP when is not positive recurrent. Theorem 9 shows that if the MBP is critical or subcritical, then there is a family of constants as such that converges in law. The limit is a standard exponential law in the critical case and a finite mixture of Erlang laws in the subcritical case. In the subcritical case, convergence in law can be strengthened to almost sure convergence (Theorem 10). Theorems 11 and 12 are central limit analogues for this almost sure convergence. Theorems 9–11 are parallel to known results for the dual version of the simple branching process in [3]. The motivation for the model studied in that reference is the fact that the simple branching process in which immigration can occur is stochastically monotone; see in [3]. Although it is not explicit in this reference, the model there is the Siegmund dual of a nonconservative or killed version of the simple branching process. This killed process is studied in [4]. The proof of almost sure convergence in Section 5 for the DMBP is quite different to that for the discrete-time version in [3]. That proof employs a general result about Markov chains for which no known continuous-time analogue is known. We remark that the dual MBP with immigration will be discussed in another paper.

Section 2 is devoted to consolidating the scattered literature concerning construction of the MBP and uniqueness of solutions of its Kolmogorov differential equation systems. In addition notation used in the sequel is established. In Section 3 we supplement the discussion of the definitions of the DMBP in [2]. In particular, Proposition 2 gives a very direct proof that is a minimal transition function. This is in contrast to the observation in [2] that this result follows from a more specialised criterion in [5].

The presentation in this paper stresses the relation with branching processes, whereas that in [2] is closer to the analytical approach used more commonly in literature on Markov construction theory. Our approach results in a more intrinsic development and, we believe, the various results have greater intuitive appeal when expressed in terms used in the branching process literature.

2. Defining and Constructing the MBP

Perusing the standard monograph accounts of branching processes reveals a variety of definitions of the MBP, all of which are motivated by the idea of the MBP as a model of a population of reproducing individuals. The branching property (1) expresses the notion of independence of separate lines of descent, and we begin by summarizing its consequences. Define the probability generating functions , and let . The branching property is equivalent to the relation . This identity implies that state 0 is absorbing. If then that is, is Feller. This property has the important consequence that all states are stable, meaning that the jump rates are all finite; see p. 43 in [6] for the definition and its consequences. Let , assumed positive, and let for , and . It follows from (1) that -matrix is specified byAssume and , where , and if and only if is conservative. Let and . We need the following notation for the sequel. Let () if and if . If denotes the least nonnegative solution of , then if , if , and if and only if (which implies that ).

Often the form (4) of -matrix is chosen to model a population process where individuals live and reproduce independently with lifetime law and, at the time of death, they are replaced by individuals with probability () or by infinitely many with probability . Thus is the mean per capita number of offspring, and we delineate the subcritical, critical, and super-critical cases, according to , , or , respectively. Then is the population size at time . If then the state-space is extended to , where is an absorbing state. In this case sample paths either hit the zero state or hit via a single infinite jump at the first occurrence of an infinite litter. In all cases, the former occurs with probability if .

The population image is captured by the minimal transition function corresponding to . This satisfies the backward and the forward Kolmogorov equations. General theory shows that is minimal because it possesses the Feller property; see p. 81 in [6]. We show this below without appealing to the Feller property. The forward equation system can be wrapped up into the single linear first-order partial differential equation:where () represents the probability generating functions for any -function. This equation has a unique solution in the class of functions which are holomorphic in the open unit disc; see p. 119 in [7]. The proof in this reference is based on a general uniqueness theorem for linear partial differential equations and although it assumes that is conservative, the proof is valid in general. Alternatively, a unique solution of the forward system within the class of transition functions follows from Reuter’s criterion that such uniqueness is equivalent to the assertion that the only sequence such that for some (and hence all) is . The explicit form of this equation isIf then setting shows that , so by recursion, . If it is clear that, given a value of , this system can be solved recursively for . So a nonnegative solution certainly exists. Estimates showing that are derived in [8].

The same end is achieved in [6] (p. 114) using a generating function approach. The proof there rather obscures the fact that (7) does have a solution. A tidier version is to formally define the generating function and observe that (7) yields the differential equation with solution If and then the integrand has a power series expansion with nonnegative coefficients. It follows that is holomorphic in the disc and, hence, if , its coefficients solve (7). However, , so .

Consequently, the minimal -function is the only transition function solution of the forward system. But if (5) holds with , then it is solved for general by . It follows immediately from Harris’ version of the uniqueness theorem that this can be the only solution holomorphic in the unit disc, and hence the minimal -function has the branching property.

It does not seem possible to so directly draw this conclusion from the uniqueness given by Reuter’s criterion. However, the branching property implies that the Chapman-Kolmogorov equations for can be rendered as the functional equation and hence that where the prime denotes partial differentiation with respect to . Setting , it follows from (4) that the first factor on the right-hand side is , and hence solves (5) with . It follows that solves the forward system, and we conclude that is the minimal transition function.

The backward system is uniquely satisfied by if it is honest. If is conservative and is dishonest, then the backward system has uncountably many transition function solutions. The single entrance solutions are investigated in [9]. Of course, none of these has the branching property.

Surprisingly, if is not conservative then the backward system is uniquely solved by the minimal transition function. This follows from another Reuter criterion which asserts that uniqueness within the class of -functions is equivalent to the condition that the only solution of the system for some (and hence all) is . This criterion is shown to hold in [8] (pp. 224–226). The following proof is simpler, though preserving the spirit of [8].

So, assume that there is a nontrivial solution , as above. Note first that because . If , then the explicit form of the above system can be written asSuppose there exists such that for all . Setting yields , a contradiction. It follows that there is a strictly increasing sequence of positive integers such that, for all , and . Consequently , and hence Letting yields , implying that , a contradiction, if . Hence the only bounded solution is .

Although is the unique solution of the backward and the forward systems if is not conservative, it is not the unique -function. Uniqueness in this case is equivalent to two conditions. The first is the above Reuter condition for uniqueness of the forward system, and the second is that, for any , there is a constant such that the inequality holds for all ; see p. 150 in [6]. However, the sum equals where we define , and since , we see that the left-hand side tends to zero as . Thus the criterion for uniqueness is violated, meaning there exists a -function which is not a solution of either the backward or forward systems.

The MBP is defined in [7] to be a Markov process whose transition function solves the forward system obtained from . The branching property is derived from this (Harris) definition. The MBP definition we adopt in Section  1 is given in [10] but not used there. Instead, (4) is motivated from population growth considerations and then used in a description of the Feller construction of the minimal process. The specification in [11] proceeds from a construction involving Galton-Watson trees with randomized split times. Also described in [11] is the Athreya-Karlin construction [12] which, in essence, is the Markov process whose jump chain is a left-continuous random walk and holding times in state have the exponential law. This construction is described too in [10]. The monograph [8], on Markov construction theory, uses our definition. This is the only treatment which, to our knowledge, gives any attention to the nonconservative case. Finally, we mention that, in [13] (see Section  3 there), the author clearly distinguishes between what he calls the probabilistic definition, that is, a Markov process having the branching property, and the analytic definition, that is, the above Harris definition. The proof in [13] of the equivalence of these definitions uses facts already mentioned that the branching property implies the Feller property, and this implies minimality.

The backward system for the minimal -function is expressed in terms of the probability generating function asor asin its integrated form. The total mass function satisfiesThis implies the known criterion that is honest if and only ifwhere . This appears in [7] where it is attributed to E. B. Dynkin. See the introduction of [14] for remarks on these attributions. Note that only if and certainly if is not conservative. It follows from (16) that if then , and as .

Suppose that is not regular. It is not at first apparent why the constructions of transition functions in [9] need to be conservative. The transition functions considered in that paper (see (2.11) there) have the form of the sum of plus a convolution integral. A computation shows that the derivative of this added term is proportional to . Since , we see from (14) with that , and hence the transition functions constructed in [9] are -functions only if is conservative. On the other hand, the limiting conditional theorem in [9] (p. 743) holds for the nonconservative MBP. Finally, we mention that several of the results in [15] for the nonconservative linear birth and death process carry over to the MBP.

3. On Definitions of the Dual MBP

As we mentioned in the introduction, three equivalent definitions of the DMBP are presented in [2]. Definition (i) is just Definition 1 above. Definition (ii) asserts that the DMBP is the Markov process whose -matrix has the formwhere is a sequence satisfyingIn particular . So , and if equality holds, then . Hence we always assume that to avoid a trivial situation. In this case the parameters and are related throughNote that if and only if is conservative. In addition , and the inverse relation is implying that unless there is at least one strict inequality in the chain (19).

Definition (iii) is that the DMBP is a Markov process whose transition function has the dual branching propertyand . The proof in [2] is a mostly analytic demonstration that (ii) (i) (iii) (ii). The following proof that (i) (iii) supplements the treatment in that reference.

We note first that, in the case that is conservative, the paper [16] gives an analytic proof that is stochastically monotone. A direct and completely general demonstration follows by expressing (1) as where the processes are independent copies of started with . It follows that and since the summands are nonnegative it is evident that is SM. This is valid whether or not is regular, not regular, or not even conservative. Hence Siegmund’s theorem is applicable in each of these circumstances because, if is dishonest, then the added boundary state is absorbing. Hence there is a Markov process satisfying Definition (i).

Next, define the generating function , which is finite if . The relation (2) is equivalent to the identity from which we obtain the fundamental identitySumming over all yields . This implies that ; that is, is always an honest transition function. We show below that it is a minimal transition function.

We now show that (1) implies the identity (22). If then (26) implies thatEquating the coefficients of on each side of this identity together with a little manipulation yields (22).

In the opposite direction, suppose that we start with Definition (iii). Then defining as above and we see that (22) is equivalent to (26). In addition, But (22) implies that the coefficient of is nonnegative and also that . Hence is the probability generating function of a possibly defective discrete law. It follows from this that has a power series expansion and that the coefficients possess the branching property (1), after swapping and . Since is standard, and hence . It follows that .

It remains to show that satisfy the Chapman-Kolmogorov equations. Since we have established the branching property, this will follow once we demonstrate the composition identityTo prove this, observe that the Chapman-Kolmogorov equations for yield for ,Setting yields and (29) follows from the definition of .

It follows that we have constructed comprise a standard transition function which possesses the branching property. Hence, as we have seen, it is a minimal transition function. Consequently there is a MBP whose transition function is that which we have constructed and that it is linked to via (18). But this identity is equivalent to (2), and hence Definitions (i) and (iii) are equivalent.

We have seen that the transition function is always honest. A sufficient condition for it to be minimal is the Feller property for all (p. 43 in [6]). To check this property, note that (2) implies that is SM, and this implies the existence of the limit . This limit is evaluated by applying an Abelian theorem for power series to (18) to obtain and this is zero if and only if , that is, if and only if is honest.

The fact that is always minimal is a corollary of a simple general result which we state as follows.

Proposition 2. Suppose is any SM transition function whose -matrix elements satisfy and which admits a dual transition function in the sense of (2).
(1) The dual -matrix is specified by and is conservative.
(2) Suppose that there is an integer such that if , and there exists such that is nondecreasing for . Then is regular if

Proof. Since , (33) follows by differentiation. Summing over yields for all , proving Assertion (1). For (2), it follows from the bounded increment assumption and (33) that if , then if , implying that . So if denotes the jump chain of , then almost surely. Hence , implying that for all sufficiently large . Thus (34) implies that , almost surely. It follows from the (generalized) Feller-Lundberg criterion (e.g., p. 337 in [17]) that the Feller minimal process constructed from is honest, and hence its transition function coincides with . Thus is regular.

The jump chain of the MBP is a random walk which is skip-free to the left, and hence we can take in Assertion (2) of Proposition 2. In addition, it follows from (33) and (4) that and hence (34) is satisfied. It follows that the DMBP transition function is honest and minimal. Some manipulation with (4) and (33) will show that has the form (19).

At this point we mention two extreme cases. If then the generating MBP is the linear death process and it is easy to show from (18) that It follows that the DMBP is a linear birth process with an independent immigration component in which individuals arrive at the event times of a Poisson process having rate .

If and , then the sample path has a single jump to at a time which has the law. In terms of the population model, the first reproduction event results in an infinite number of offspring. The corresponding DMBP can be regarded as a uniform catastrophe process: if , then the next jump occurs after a time having the law, and it is into a state with probability . The zero state is absorbing.

4. Long-Term Behaviour of

Recall that is the distribution function of the hitting time of state 0 by the MBP when and that . It follows from (26) thatand hence that the DMBP is positive recurrent if and only if and . In this case there is a geometric limiting-stationary law . This is an alternative approach to Theorem  3.1 in [2].

The next two theorems require the following extension of a known result (p. 115 in [10]). Still assuming that , we define the parameter . Thenwhere , if and ; otherwise, if and . Recalling (17), we see that if and if .

It follows from (18) and (20) that . Assume for the following that in which case sample paths of the DMBP are nonincreasing step functions and 0 is an absorbing state. Thus the DMBP is a kind of generalized death process, consistent with the fact that the pure (resp., linear) death process is the Siegmund dual of the pure (resp., linear) birth process. This fact follows by inspection of -matrices exhibited in Example (d) of [1]. Still assuming , it follows from (38) thatwhere and if .

Let be the hitting time of 0 by . Since 0 is accessible from all , it follows that .

Theorem 3. If then

Proof. It follows from (39) that , and hence (2) yields and (40) follows from (39). Next, and hence , and it follows from (40) that Assertion (41) follows from this estimate and (40).

Further calculation shows that the speed of convergence in (41) is characterized by

Next we let and determine the rate at which the variation distance between and the geometric limiting-stationary law approaches zero. This is equivalent to estimating the speed at which . A more stringent measure of the speed of convergence is to require that this limit relation holds uniformly in , that is, that . If this holds then is said to be strongly ergodic. As observed in [2], is strongly ergodic if and only if is dishonest. This follows from Theorem  2.2 in [18] asserting that an ergodic and SM transition function is strongly ergodic if and only if it is not Feller, and we have seen above that is Feller if and only if is honest.

The following result gives the rate of convergence to zero of and . As a by-product, it gives a proof of the result just discussed without using Theorem  2.2 in [18].

Theorem 4. Suppose and . (a) For all , .
(b) Suppose also that . Then .
These convergence rate estimates are optimal.

Proof. We obtain exact element-wise convergence rates as follows. It follows from (26) that and hence (38) yields This implies thatNote that .
Next, (26) can be expressed as from which we obtain the representationwhere andObserving that , we write and henceby virtue of (38) and (47).
Next, (2) implies thatand, hence, if , thenSumming over , it follows from (49) and (52) that But (38) implies that , thus proving Assertion (a). It follows from (47) that the exponent cannot be increased.
Returning to (53), observe that The sum is zero if , so the right-hand side decreases as increases. Consequently and we see that the right-hand side tends to zero if and only if , that is, if and only if . Hence is not strongly ergodic if .
Suppose that . Then (38) with and shows that . Using this estimate with (52) yields In a similar way, it follows from (50) that and hence . It follows that is strongly ergodic if , and Assertion (2) follows from these estimates.

We now consider the case in which case . So setting in (26) yields for all , and hence 0 is accessible from . In addition for all , and hence is irreducible. Taking in (26) yields , since , so is not positive recurrent. It follows that is transient if the mean time to extinction of the MBP and null-recurrent if . Let in the integral and observe that (14) implies that . It follows that where and hence is transient if and it is null-recurrent if . This is equivalent to Theorem  3.2 in [2]. Since as , it is obvious that is finite if ; that is, is transient. The typical case for is that the variance is finite. In this case , and we see that ; that is, is null-recurrent. It is well known that there are examples where and .

More detail about the asymptotic behaviour of can be extracted from known results about . A clue as to what can be expected is exposed by letting in (26). This yields where denotes the Malthusian parameter of the MBP. This result shows that the uniform measure on is -invariant for . See Section  5.2 in [6] for this notion. If then , and it is known that where is slowly varying at infinity and if , and otherwise. In additionwhere is a nondefective probability generating function and for all . See p. 121 in [11] for these facts. These ’s comprise -invariant measure for (restricted to ), which may be expressed in terms of generating functions asThe generating function of the tail masses is .

Theorem 5. Let . Then is -invariant function for and, as ,

Proof. Using (26) we computeby virtue of (26) with and (64), and the -invariance assertion follows. The asymptotic relation follows because (26) and (63) imply that and the extended continuity theorem then yields the assertion.

Turning to the case , we have the following result similar to Theorem 5.

Theorem 6. If then

Proof. The assertion follows once we show that , and this will follow from This is quite well known, and it can be shown easily as follows. Let and choose such that . This is possible since and is a continuous distribution function. Then , and hence the limit (69) is since .

We can obtain a second-order correction to this result by using the following further facts about the MBP. Recall that , and observe that since is a probability generating function if ; then has a Maclaurin expansion with nonnegative coefficients. Hence the function is finite if , , and . The integrated backward equation (15) can be expressed asSetting shows that is the inverse function of and that . It follows too from (71) that ; that is, is an invariant measure for restricted to . It is known to be unique up to multiplication by constants. The following proposition expresses in terms of the precise rate at which tends to zero.

Proposition 7. The limit

Proof. A Taylor expansion of about yieldsSince is a decreasing function, it follows from (15) and then (71) that and hence that If at least one extreme side of this inequality is bounded away from as through some sequence, then (73) will be violated. It follows that the limit as of each bound exists and equals . Setting , then since , we conclude that and the assertion follows.

Let and for . Then .

Theorem 8. If and , then

Proof. The generating function of the term in the numerator is Dividing by and letting , it follows from Proposition 7 that the first term in the right-hand side numerator contributes the limit . The second term in that numerator is asymptotically equal to . Choosing such that , we find that so a double application of L’Hopitâl’s rule shows that the limit of the right-hand side, as , equals since , and . See p. 109 in [10] for the moment formulae. The assertion follows from these limit identities.

If then it can be shown using (33) that , thus giving a precise estimate of the second order correction supplied by Theorem 8.

5. Limit Theorems

In this section we assume that and, if , then we shall understand (63) to hold with . Our first result shows that converges in law. We denote this mode of convergence by .

Theorem 9. (a) If then, for all , , where has the standard exponential law.
(b) If and , then , where the law of is a finite mixture of Erlang laws, having the density function (85) below.

Proof. It follows from (26) that Replacing with , it follows from (63) thatAssertion (a) follows from the extended continuity theorem for probability generating functions and is the Laplace-Stieltjes transform of the standard exponential law.
If then the right-hand side of (81) can be expanded in the form But where is the one-step transition probability of the Markov chain which is the (modified) Siegmund dual of the simple branching process whose offspring probability generating function is . This offspring law is supercritical, and, indeed, since , it follows that if . See [3] for properties of this dual. We conclude from (81) thatand hence that Assertion (b) holds, where has the Laplace-Stieltjes transform on the right-hand side of (84), and its density function isThis completes the proof.

We show next that the the mode of convergence in Theorem 9(b) can be strengthened to almost sure convergence. Our proof exploits the upward skip-free nature of DMBP sample paths. It follows quite closely the progression through Theorems  2.2 to  2.4 in [19], which in turn is an application of basic methodology developed in [20].

Theorem 10. If and , then .

Proof. Let be the hitting time of state by the DMBP. The skip-free property of its sample paths implies that if then , where is the first-passage time from to , and ’s are independent. Writing , the quantities are computed by solving the linear systemIt is shown in [20] that this system has the stated solution and it is unique up to constant factors. Indeed, , and starting with , the system can be solved recursively for .
Formally define the generating function . Recalling that , it follows from (4) and (33) that which together with (87) yields the differential equation Integration yields the solution where, we recall, . The right-hand side has a Maclaurin expansion whose coefficients are positive and which satisfies system (87). The function defined at (63) is related to bywhere is slowly varying at infinity; see p. 122 in [11] for the right-hand side representation. It follows that Since is almost surely nondecreasing in , then is nondecreasing and hence a Tauberian theorem for power series yields the asymptotic equivalencein other words, as we have the asymptotic equivalenceFor , define and observe that (91) takes the form It follows that (94) can be expressed as The right-hand side is the (bilateral) Laplace-Stieltjes transform of a random variable which has the Gumbel type distribution function . We have thus shown that . But the left-hand side is a sum of independent random variables, so it follows from the equivalence theorem for convergent random series of independent summands that .
Recalling that , it follows that, -almost surely, where and and are independent. Hence the Laplace-Stieltjes transform of the limit isNext, let . Then is a positive-valued Lévy martingale, and . On the other hand,where we have used the strong Markov property for the second equality, (97) and the dominated convergence theorem for the penultimate equality and then (98). Letting in this result, (93) and (98) show that if , then almost surely that is, , -almost surely. This conclusion can be expressed as the asymptotic equivalence . Since is regularly varying with index , its inverse is regularly varying with index , and hence The assertion follows after some algebra to check that the definitions of , , and imply that .

If then the condition is equivalent toAssuming this, it follows thatConsequently, the norming in Theorem 9 has the asymptotic form . In addition, as , and hence the norming function emerging in the final stages of the previous proof satisfies . These equivalences show that the limit laws in Theorems 9 and 10 agree, and we have the identification -almost surely.

We will state and prove two results concerning the rate at which converges to . First we recollect some known results concerning defined by (63). If we let in (64), then differentiating with respect to and recalling (14) yieldIntegration leads to the identity It follows that the mean of the limiting conditional law for the MBP is . We will assume that is finite, in which case differentiating (105) leading to is finite. Hence we have the expansionFinally, we let .

Theorem 11. If and , then where the law of is the finite mixture of generalized Laplace laws whose characteristic function is

Proof. First we derive the Laplace-Stieltjes transform of . The Markov property implies thatLet , and suppose that is restricted to imaginary values. Multiplying through by and using the dominated convergence theorem we obtain the identityIt follows from (26) and (84) that Next, replace with . Since as , the factorUsing (64) with and then (108), we haveExpanding gives So it follows from (108) that Adding this to (115) cancels terms in . It follows from (113), (26) with , and (114) thatWriting , where is real-valued, it follows as in the proof of Theorem 9 that the characteristic function of converges to , as asserted. This completes the proof.

The term in (110) is the characteristic function of the sum of independent copies of , where has a standard Laplace law. If are independent copies of , then , where ’s are independent and have the Erlang density for . It follows that the density of is where is Bateman’s function, and the evaluation is Entry 3.38.8 in Gradshteyn and Ryzhik [21]. Hence the limit law has the density function , which is a Laplace law density function if .

Let have a standard normal law and be independent of the above ’s. Then the characteristic function of is , and hence , where is as defined in Theorem 9(b), and it is independent of . It follows that it ought to be possible to simplify the mixed limit law in Theorem 11 by replacing the norming function with . The following result achieves this. To see why, let and observe that Theorem 11 can be expressed as . Replacing in the norming by will eliminate this factor from the weak limit.

Recall from the end of the last proof that, under the second-order moment assumption assumed in the following result, .

Theorem 12. If and , then

Proof. We begin by refining this remainder term in the expression for which we deduced from (104). The second moment condition allows us to write so Consequently and henceFor let and observe that , and hence It follows that if , then ; that is,We prove asymptotic normality of the right-hand side as follows. We find from (98) thatwhere has a standard Gumbel law. Inverting the Laplace-Stieltjes transforms gives the identitysince , and we have used (91). The exponential factor in (129) is the probability generating function of a random variable, say, which has a compound Poisson law. We thus obtain the representation Let , and denote the random infinite series in (127) by . The representation then takes the form and we evaluate the limit of the right-hand side as follows.
Noting that , and recalling that , we find from (125) thatIt follows from (108) thatThis implies that , and hence that Thus we have shown that Theorem  4.1 in [20] asserts that this is equivalent to the following assertion: if , then Using (125), we write this more explicitly as We have seen, under the second-order moment condition, that Theorem 10 can be expressed as (-almost surely). So Hence as asserted.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.