Almost Sure Limit Theorems for Sums and Maxima from the Domain of Geometric Partial Attraction of Semistable Laws

István Berkes ^{1} Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences P.O.B. 127, H–1364 Budapest, Hungary

Endre Csáki ^{2} Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences P.O.B. 127, H–1364 Budapest, Hungary

Sándor Csörgő ^{3} Bolyai Institute, University of Szeged Aradi vértanúk tere 1, H-6720 Szeged, Hungary and Department of Statistics, University of Michigan 4062 Frieze Building, Ann Arbor, MI 48109–1285, U.S.A.

Zoltán Megyesi ^{4} Bolyai Institute, University of Szeged Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Dedicated to Pál Révész for his sixty-fifth birthday

Abstract. The possible limiting distributions of sums of independent identically distributed random variables along subsequences {k_{n}} ⊂ ℕ satisfying k_{n+1}/k_{n} → c ≥ 1 are the semistable laws and the domain of geometric partial attraction of a semistable law consists of distributions attracted to it along such a subsequence. The aim of this paper is to show that sums and maxima from the domain of geometric partial attraction of a semistable law satisfy almost sure limit theorems along the whole sequence {n} = ℕ of natural numbers, despite the fact that ordinary convergence in distribution typically takes place in both cases only along {k_{n}} and related subsequences. We describe the class of all possible almost sure asymptotic distributions both for sums and maxima.

AMS 1991 Subject Classification: Primary 60F15; Secondary 60E07. Keywords: Semistable laws, domains of geometric partial attraction, sums and maxima, asymptotic distributions, logarithmic averages, almost sure limit theorems.

1,2 Research supported by the Hungarian National Foundation for Scientific Research, Grants T 19346 and T 29621.

3 Research supported in part by the NSF Grant DMS–9625732 held at the University of Michigan and by the Hungarian National Foundation for Scientific Research, Grants T 032025 and T 034121.

4 Research partially supported by the ‘Students for Science’ program of the Pro Renovanda Cultura Hungariae Foundation and by the Hungarian National Foundation for Scientific Research, Grants T 032025 and T 034121.

1. Introduction and results: natural norming sequences

Let X_{1}, X_{2}, … be independent random variables with partial sums S_{n} = X_{1} + ⋯ + X_{n}, n ∈ ℕ = {1, 2, …}. Introduce the symbol C_{G} for the continuity points of a function G. In the last decade, many authors investigated the connection between the weak limit theorem

ℙ\{\frac{S_{n} − b_{n}}{a_{n}} ≤ x\} → G(x),  for any x ∈ C_{G},  (A)

and, abbreviating ‘almost surely’ or ‘almost sure’ to ‘a.s.’, the corresponding pointwise result

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{S_{r} − b_{r}}{a_{r}} ≤ x\} → G(x),  a.s. for any x ∈ C_{G},  (B)

where a_{n} > 0 and b_{n} are some numerical sequences and I denotes the indicator function; throughout, we use the convention that every asymptotic relation is meant as n → ∞ unless otherwise specified.

Berkes and Dehling [6] proved the surprising result that under mild moment conditions on the X_{n}, relation (A) implies relation (B), despite the almost sure nature of the latter. If the random variables X_{1}, X_{2}, … are identically distributed, then the limiting distribution in (A) is necessarily stable with some exponent α ∈ (0, 2 ], and the theorem of Berkes and Dehling yields the well-known results of Brosamler [8], Schatte [19], Lacey and Philipp [14] and Fisher [12] for (B) when 𝔼(X_{1}^{2}) < ∞, and even the rest of the case α = 2, and that of Peligrad and Révész [18] in the case 0 < α < 2. For a recent survey of results in the area of “logarithmic” limit theorems of type (B), see Berkes [2].

In the converse direction, Berkes, Dehling and Móri [7] constructed examples for which (B) holds while (A) does not (and Móri [17] discovered interesting natural examples, different from sums, with the same feature). Berkes, Csáki and Csörgő [5] recently showed that this amusing phenomenon for sums of independent and identically distributed random variables obtains not just for esoteric examples, but also in a well-known classical situation: for the underlying distribution of the St. Petersburg game the almost sure statement in (B) holds with a suitable G, despite the fact that convergence in (A) does not take place along the whole {n} = ℕ, and the same is true for the maximal gains in a sequence of games. Our aim here is to extend these results and thus to show that the phenomenon holds not just for isolated examples, but in fact remains true for a whole class of distributions, namely, for those in the domain of geometric partial attraction of semistable laws. To state the results we need some notation and preliminary facts.

The concept of a ‘semistable’ distribution first appeared in 1937 in Paul Lévy’s fundamental work [15]. We shall use recent results of Megyesi [16], where the existing theory of semistable laws is inserted into the framework of the ‘probabilistic’ approach of Csörgő, Haeusler and Mason [10] and Csörgő [9]. For more results and further references see [16]. To start with standard notions and notation used in this approach, let Ψ be the class of all non-positive, non-decreasing, right-continuous functions ψ(·) defined on the positive half-line (0, ∞) such that the integrals ∫_{ε}^{∞} ψ^{2}(s) ds < ∞ for all ε > 0. Let E_{1}^{(j)}, E_{2}^{(j)}, …, j = 1, 2, be two independent sequences of independent exponentially distributed random variables with mean 1, and with respective partial sums Y_{n}^{(j)} = E_{1}^{(j)} + ⋯ + E_{n}^{(j)} as jump points, and consider the standard left-continuous independent Poisson processes N_{j}(u) = ∑_{n=1}^{∞} I{Y_{n}^{(j)} < u}, 0 ≤ u < ∞, j = 1, 2. For a function ψ ∈ Ψ, define the random variables

W_{j}(ψ) = ∫_{1}^{∞}[N_{j}(s) − s]dψ(s) + ∫_{0}^{1}N_{j}(s) dψ(s) − ψ(1), j = 1, 2 ,

where the first integral is almost surely well-defined, by the condition that ψ ∈ Ψ, as an improper Riemann integral. Let Z be a standard normal random variable such that N_{1}(·), Z, and N_{2}(·) are independent, and for ψ_{1} ∈ Ψ and ψ_{2} ∈ Ψ and a finite constant σ ≥ 0 finally introduce the random variables

V (ψ_{1}, ψ_{2}, σ) = −W_{1}(ψ_{1}) + σZ + W_{2}(ψ_{2}).  (1.1)

Considering then the constant, also for ψ ∈ Ψ,

Θ(ψ) = ∫_{0}^{1} \frac{ψ(s)}{1 + ψ^{2}(s)} ds − ∫_{1}^{∞} \frac{ψ^{3}(s)}{1 + ψ^{2}(s)} ds

and letting W (ψ_{1}, ψ_{2}, σ) = V (ψ_{1}, ψ_{2}, σ) + Θ(ψ_{2}) − Θ(ψ_{1}), by Theorem 3 in [10] this random variable has characteristic function given for all t ∈ ℝ as

\begin{aligned}𝔼(e^{itW(ψ_{1},ψ_{2},σ)}) = exp\{−\frac{σ^{2}}{2} t^{2} & + ∫_{−∞}^{0}(e^{itx} − 1 − \frac{itx}{1 + x^{2}} )dL(x) \\ & + ∫_{0}^{∞}(e^{itx} − 1 − \frac{itx}{1 + x^{2}} )dR(x)\}, \end{aligned}  (1.2)

where L(x) = inf{s > 0 : ψ_{1}(s) ≥ x}, x < 0, and R(x) = − inf{s > 0 : ψ_{2}(s) ≥ −x}, x > 0. Here L(·) is left-continuous and non-decreasing on (−∞, 0) with L(−∞) = 0 and R(·) is right-continuous and non-decreasing on (0, ∞) with R(∞) = 0, and we have ∫_{−ε}^{0} x^{2}dL(x) + ∫_{0}^{ε} x^{2}dR(x) < ∞ for every ε > 0 since ψ_{1}, ψ_{2} ∈ Ψ. Thus V (ψ_{1}, ψ_{2}, σ) is infinitely divisible by Lévy’s formula; see [13] for example. Conversely, given the right side of (1.2) with L(·) and R(·) having the properties just listed, the variable W (ψ_{1}, ψ_{2}, σ) has this characteristic function again if we choose ψ_{1}(s) = inf{x < 0 : L(x) > s} and ψ_{2}(s) = inf{x<0: −R(−x)>s}, s > 0, for then ψ_{1}, ψ_{2} ∈ Ψ.

Thus, modulo a location constant, the class I of all non-degenerate infinitely divisible distributions can be identified with the class {(ψ_{1}, ψ_{2}, σ) ≠ (0, 0, 0) : ψ_{1}, ψ_{2} ∈ Ψ, σ ≥ 0} of triplets. We say that F is in the domain of partial attraction of a G = G_{ψ_{1},ψ_{2},σ} ∈ I if there exists a subsequence {k_{n}}_{n=0}^{∞} ⊂ ℕ and centering and norming constants B_{k_{n}} ∈ ℝ and A_{k_{n}} > 0 such that

\frac{1}{A_{k_{n}}} \{∑_{j=1}^{k_{n}} X_{j} − B_{k_{n}} \} ⟶^{D} V (ψ_{1}, ψ_{2}, σ),  (1.3)

where X_{1}, X_{2}, … are independent identically distributed random variables with common distribution function F (x) = ℙ(X_{1} ≤ x), x ∈ ℝ, the symbol ⟶^{D} denotes convergence in distribution and G_{ψ_{1},ψ_{2},σ} is the distribution function of the random variable V (ψ_{1}, ψ_{2}, σ). All subsequences of ℕ appearing in this paper are assumed to be unbounded. If we demand {k_{n}} = {n} = ℕ in (1.3) then the possible limiting distributions are the stable laws and F is said to be in the domain of attraction of the stable law in question, in which case either (ψ_{1}, ψ_{2}, σ) = (0, 0, σ) for some σ > 0, and F is in the domain of attraction of the normal distribution, written F ∈ 𝔻(2), or (ψ_{1}, ψ_{2}, σ) = (m_{1}ψ^{α}, m_{2}ψ^{α}, 0) for some constants α ∈ (0, 2), m_{1}, m_{2} ≥ 0, m_{1} + m_{2} > 0, where ψ^{α}(s) = −s^{−1/α}, s > 0, in which case F is in the domain of attraction of a stable distribution of exponent α, written F ∈ 𝔻(α). (The superscript α in ψ^{α} here, and in ψ_{1}^{α} and ψ_{2}^{α} beginning with (1.5) below, is meant as a label, not as a power exponent.) The normal being the stable law of exponent 2, let S denote the class of all stable laws.

The class S_{∗} ⊂ I of semistable laws arises as the class of limiting distributions in (1.3) if we place a geometric growth condition on the subsequence {k_{n}}, i.e., that

k_{n+1}/k_{n} → c for some c ∈ [ 1, ∞).  (1.4)

By Theorem 1 of [16], G_{ψ_{1},ψ_{2},σ} ∈ S_{∗} if and only if either (ψ_{1}, ψ_{2}, σ) = (0, 0, σ) for some σ > 0, giving the normal distribution as a semistable distribution of exponent 2, or (ψ_{1}, ψ_{2}, σ) = (ψ_{1}^{α}, ψ_{2}^{α}, 0), where

ψ_{j}^{α}(s) = M_{j}(s)ψ^{α}(s) = −M_{j}(s)s^{−1/α}, s > 0, j = 1, 2 ,  (1.5)

for some α ∈ (0, 2), defining a semistable law of exponent α ∈ (0, 2), where M_{1} and M_{2} are non-negative, right-continuous functions on (0, ∞), either identically zero or bounded away from both zero and infinity, such that M_{1} +M_{2} is not identically zero, the functions M_{j}(·)ψ^{α}(·) are non-decreasing and, if c > 1 in (1.4), then M_{j}(cs) = M_{j}(s) for all s > 0, j = 1, 2. This property will be referred to as multiplicative periodicity with period c. If c = 1 in (1.4), then the functions M_{1} and M_{2} are necessarily constant, giving the stable special case of semistable laws, therefore S ⊂ S_{∗}. For α ∈ (0, 2), Lévy’s original description of property (1.5) in terms of L and R in (1.2) is that there exist non-negative bounded functions M_{L}(·) on (−∞, 0) and M_{R}(·) on (0, ∞), one of which has a strictly positive infimum and the other one either has a strictly positive infimum or is identically zero, such that L(x) = M_{L}(x)/|x|^{α}, x < 0, is left-continuous and non-decreasing on (−∞, 0) and R(x) = −M_{R}(x)/x^{α}, x > 0, is right-continuous and non-decreasing on (0, ∞), while M_{L}(c^{1/α} x) = M_{L}(x) for all x < 0 and M_{R}(c^{1/α} x) = M_{R}(x) for all x > 0, for the same period c > 1. Because of the inversions given above, the two descriptions are equivalent.

Now, for G = G_{ψ_{1},ψ_{2},σ} ∈ S_{∗}, we say that F is in the domain of geometric partial attraction of G with rank c ≥ 1, in short F ∈ 𝔻_{gp}^{(c)}(G), if (1.3) holds along a subsequence {k_{n}} ⊂ ℕ satisfying the growth condition (1.4) if c > 1. Recalling that (ψ_{1}, ψ_{2}, σ) ≠ (0, 0, 0) for G = G_{ψ_{1},ψ_{2},σ} ∈ S_{∗}, define c = c(G_{0,0,σ}) = 1 for any σ > 0 and c = c(G_{ψ_{1}^{α},ψ_{2}^{α},0}) = inf{c > 1 : M_{j}(cs) = M_{j}(s), s > 0, j = 1, 2}, the minimal common period c (among those greater than 1) of the factor functions M_{1} and M_{2} in ψ_{1}^{α} and ψ_{2}^{α} in (1.5) for α ∈ (0, 2). It turns out for the whole domain 𝔻_{gp}(G) = ⋃_{c≥1} 𝔻_{gp}^{(c)}(G) of geometric partial attraction of G ∈ S_{∗} that 𝔻_{gp}(G) = ⋂_{m∈ℕ} 𝔻_{gp}^{(c^{m})}(G) = 𝔻_{gp}^{(c)}(G). Also, if c(G) = 1 for G ∈ S_{∗}, then G ∈ S and 𝔻_{gp}(G) = 𝔻(G), the domain of attraction of the stable G. In other words, if 𝔻(S) = ⋃_{G∈S} 𝔻(G) = ⋃_{0<α≤2} 𝔻(α) is the classical domain of attraction and 𝔻_{gp}(G) = ⋃_{G∈G} 𝔻_{gp}(G) is the domain of geometric partial attraction of a class G ⊂ S_{∗}, then 𝔻_{gp}(S) = 𝔻(S).

A characterization of an F ∈ 𝔻_{gp}(S_{∗}) in terms of the quantile function

Q(s) = inf{x ∈ ℝ: F (x) ≥ s}, 0 < s < 1,

is given by Theorem 3 in [16]. Consider a subsequence {k_{n}}_{n=0}^{∞} ⊂ ℕ satisfying (1.4). If c = 1 in (1.4), then put γ(s) = 1, 0 < s < 1. If c > 1, then the sequence {k_{n}} is eventually strictly increasing, thus for all s ∈ (0, s_{0}), with s_{0} ∈ (0, 1] small enough, there exists a unique k_{n^{∗}(s)} such that k_{n^{∗}(s)}^{−1} ≤ s < k_{n^{∗}(s)−1}^{−1} . We define γ(s) = sk_{n^{∗}(s)} for s ∈ (0, s_{0}) and γ(s) = 1 for s ∈ [s_{0}, 1), so that 1 ≤ γ(s) < c + ε for any fixed ε > 0 and all s ∈ (0, 1) for the limiting c ≥ 1 from (1.4). Let Q_{+}(·) denote the right-continuous version of the quantile function Q(·). Since 𝔻_{gp}(G) = 𝔻(G) for a normal G ∈ S_{∗}, we only have to describe the domain of geometric partial attraction of non-normal semistable laws, for which the characterization is the following: If G_{ψ_{1}^{α},ψ_{2}^{α},0} ∈ S_{∗} is semistable with exponent α ∈ (0, 2), so that ψ_{1}^{α} and ψ_{2}^{α} satisfy (1.5), and F ∈ 𝔻_{gp}(G_{ψ_{1}^{α},ψ_{2}^{α},0}) so that (1.3) hold for V (ψ_{1}^{α}, ψ_{2}^{α}, 0) and a subsequence {k_{n}}_{n=0}^{∞} ⊂ ℕ satisfying (1.4), then for all s ∈ (0, 1),

\begin{aligned}& Q_{+}(s) = −s^{−1/α}ℓ(s)[M_{1}(γ(s)) + h_{1}(s)] \\ and \\ & Q(1 − s) = s^{−1/α}ℓ(s)[M_{2}(γ(s)) + h_{2}(s)] \end{aligned}  (1.6)

for the same α ∈ (0, 2), where ℓ(·) is a right-continuous function, slowly varying at zero, the function γ(·) is determined by the subsequence {k_{n}} along which (1.3) holds and the errors h_{1} and h_{2} are right-continuous functions such that if M_{j} is continuous, then lim_{s↓0} h_{j}(s) = 0 for the corresponding h_{j}, while if M_{j} has discontinuities, then the corresponding h_{j}(s) may not go to zero as s ↓ 0 but lim_{n→∞} h_{j}(t/k_{n}) = 0 for every continuity point t > 0 of M_{j}, j = 1, 2. Conversely, if for the quantile function pertaining to F the equations in (1.6) hold with the properties of ℓ and of h_{1} and h_{2} just listed, for some α ∈ (0, 2) and functions M_{1} and M_{2} satisfying the properties described at (1.5), and for γ(·) determined by a given subsequence {k_{n}}_{n=0}^{∞} ⊂ ℕ satisfying (1.4), then F ∈ 𝔻_{gp}(G_{ψ_{1}^{α},ψ_{2}^{α},0}) for the ψ_{1}^{α} and ψ_{2}^{α} given by (1.5), and, in particular, the relation (1.3) can be specified as

\frac{1}{k_{n}^{1/α}ℓ(1/k_{n})}\{∑_{j=1}^{k_{n}} X_{j} − k_{n}∫ 1− \frac{1}{kn}_{\frac{1}{kn}} Q(u) du\} ⟶^{D} V (ψ_{1}^{α}, ψ_{2}^{α}, 0).  (1.7)

To state the main results from [11], which will form the basis of our further investigations, for λ > 0 introduce the functions

ψ_{j}^{λ,α}(s) := λ^{−1/α}ψ_{j}^{α}(s/λ) = −M_{j}(s/λ)s^{−1/α}, s ∈ (0, ∞),

j = 1, 2, and the distribution function

G_{λ}(x) := ℙ{V (ψ_{1}^{λ,α}, ψ_{2}^{λ,α}, 0) ≤ x}, x ∈ ℝ.

Clearly, G_{ψ_{1}^{α},ψ_{2}^{α},0} = G_{1} = G_{c^{n}} for all n ∈ ℕ and G_{λ} is semistable for all λ > 0. Let now F ∈ 𝔻_{gp}(G_{ψ_{1}^{α},ψ_{2}^{α},0}) along a subsequence {k_{n}} satisfying (1.4), in which there is no loss of generality to assume that c > 1. Introduce now

γ_{n} := \frac{1}{γ(1/n)} = \frac{n}{min\{k_{m} : k_{m} ≥ n\}}, n ∈ ℕ.

Again, it is easy to see that γ_{n} ≤ 1 and lim inf_{n→∞} γ_{n} = 1/c for the c figuring in (1.4). Now the Merge Theorem is as follows:

sup_{x∈ℝ} |ℙ\{\frac{∑_{j=1}^{n} X_{j} − n∫ 1− \frac{1}{n}_{\frac{1}{n}} Q(u) du}{n^{1/α}ℓ(1/n)} ≤ x\} − G_{γ_{n}}(x)| → 0.  (1.8)

In fact, Theorem 2 in [11] states more: introducing X_{1,n} ≤ X_{2,n} ≤ ⋯ ≤ X_{n,n}, the order statistics pertaining to X_{1}, X_{2}, … X_{n}, it states that the suitably centered and normalized trimmed sums ∑_{j=l}^{n−m} X_{j,n}, where l and m are some fixed integers, merge together with ‘trimmed variants’ of G_{γ_{n}}. The above statement corresponds to the untrimmed special case, i.e., when l = m = 0. Our first theorem is now the following:

Theorem 1. Assume that F ∈ 𝔻_{gp}(G_{ψ_{1}^{α},ψ_{2}^{α},0}) along a subsequence {k_{n}} satisfying (1.4) with some c > 1. Then

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{∑_{j=1}^{r} X_{j} − r∫ 1− \frac{1}{r}_{\frac{1}{r}} Q(u) du}{r^{1/α}ℓ(1/r)} ≤ x\} → \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ}(x)}{γ} dγ  (1.9)

almost surely for all x ∈ ℝ.

Turning now to maxima, for two distribution functions G and H introduce

L(G, H) = inf{ε > 0 : G(x − ε) − ε ≤ H(x) ≤ G(x + ε) + ε for all x ∈ ℝ},

their Lévy distance. It is well-known that this distance metrizes the weak convergence of distribution functions on the line (cf. p. 33 in [13]). Recalling our previous notation for order statistics and the ingredients of the Lévy measure of non-normal semistable laws, introduce

H_{n}(x) := ℙ\{\frac{X_{n,n}}{n^{1/α}ℓ(1/n)} ≤ x\} , x ∈ ℝ,

and for λ > 0,

K_{λ}(x) := \{ \begin{matrix}0, & x ≤ 0, \\ e^{−M_{R}(λ^{1/α}x)x^{−α}}, & x > 0. \end{matrix}  (1.10)

Here K_{λ} can be identified as the distribution function of |ψ_{2}^{λ,α}(Y)|, where Y is an exponentially distributed random variable with mean 1. When considering maxima, we will assume that M_{2} ≢ 0, and hence M_{R} ≢ 0. Then, retaining the conditions of Theorem 1, the Merge Theorem for Maxima (Theorem 3 in [11]) states that

L(H_{n}, K_{γ_{n}}) → 0.  (1.11)

The merge of distribution functions, unlike in the case of sums, does not generally take place uniformly, that is, it may not hold in the supremum distance. The main reason for this is that the distribution functions K_{λ} are not continuous if M_{R} is not. The almost sure limit theorem is nevertheless true:

Theorem 2. Assume that F ∈ 𝔻_{gp}(G_{ψ_{1}^{α},ψ_{2}^{α},0}) along the subsequence {k_{n}} satisfying (1.4) with some c > 1, where ψ_{2}^{α} ≢ 0. Then

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{X_{r,r}}{r^{1/α}ℓ(1/r)} ≤ x\} → \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{K_{γ}(x)}{γ} dγ  (1.12)

almost surely for all x ∈ ℝ.

We note that, even though K_{γ}(·) may not be continuous, the limiting distribution function in (1.12) is continuous at every x ∈ ℝ, as will be shown in the proof.

The proofs of these results are in the next section, after which Section 3 is devoted to the scaling problem: what happens if, instead of r^{1/α}ℓ(1/r) in the two theorems above, one uses a general norming sequence?

2. Proof of Theorems 1 and 2

Lemma 1. Let {X_{n}} be a sequence of independent identically distributed random variables with partial sums S_{n} and assume that for some numerical sequences {α_{n}} and {β_{n}}, satisfying

C_{1}(l/k)^{λ} ≤ α_{l}/α_{k} ≤ C_{2}(l/k)^{λ}  for any l > k,  (2.1)

for some λ ∈ [ 1/2, ∞), where C_{1} and C_{2} are finite positive constants, the sequence

\{\frac{S_{n} − β_{n}}{α_{n}} \}_{n=1}^{∞}  (2.2)

is bounded in probability. Then

sup_{n} 𝔼(|\frac{S_{n} − β_{n}}{α_{n}} |^{p}) < ∞

for any 0 < p < 1/λ.

Proof. The proof of this lemma is implicit in the proof of Theorem 6.1 in de Acosta and Giné [1].

Proof of Theorem 1. Introduce A_{n} := n^{1/α}ℓ(1/n) and B_{n} := n∫ 1− \frac{1}{n}_{\frac{1}{n}} Q(u) du for the normalizing and centering sequences in (1.8) and (1.9), respectively. First we prove that

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} ℙ\{\frac{1}{A_{r}} [∑_{j=1}^{r} X_{j} − B_{r}] ≤ x\} → \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ}(x)}{γ} dγ,  (2.3)

for all x ∈ ℝ. By the Merge Theorem in (1.8) it suffices to show that

\frac{1}{log n} ∑_{r=1}^{n} \frac{G_{γ_{r}}(x)}{r} → \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ}(x)}{γ} dγ,  (2.4)

for all x ∈ ℝ. First we infer that (2.4) holds along {k_{n}}. Indeed,

\frac{1}{log k_{n}} ∑_{r=1}^{k_{n}} \frac{G_{γ_{r}}(x)}{r} = \frac{1}{log k_{n}} ∑_{r=1}^{k_{0}} \frac{G_{γ_{r}}(x)}{r} + \frac{1}{log k_{n}} ∑_{m=1}^{n} ∑_{r=k_{m−1}+1}^{k_{m}} \frac{G_{γ_{r}}(x)}{r}

= o(1) + \frac{1}{log k_{n}} ∑_{m=1}^{n} ∑_{r=k_{m−1}+1}^{k_{m}} \frac{G_{r/k_{m}}(x)}{r/k_{m}} \frac{1}{k_{m}}

= o(1) + \frac{n log c}{log k_{n}} \frac{1}{log c} \frac{1}{n} ∑_{m=1}^{n} ∑_{r=k_{m−1}+1}^{k_{m}} \frac{G_{r/k_{m}}(x)}{r/k_{m}} \frac{1}{k_{m}}

Here the second equality follows by the fact that γ_{r} = r/k_{m} for r ∈ {k_{m−1} + 1, …, k_{m}}. Introduce

\overline{G}_{m}(s, x) := \{ \begin{matrix}\frac{G_{j/km} (x)}{j/k_{m}} , & s ∈ (\frac{j}{k_{m}} , \frac{j+1}{k_{m}}] , j = k_{m−1} + 1, …, k_{m}, \\ 0 , & elsewhere. \end{matrix}

If μ ↑ λ ∈ (0, ∞), then, by the right-continuity of ψ_{1}^{α}(·) and ψ_{2}^{α}(·),
V (ψ_{1}^{μ, α}, ψ_{2}^{μ, α}, 0) → V (ψ_{1}^{λ,α}, ψ_{2}^{λ,α}, 0),

even almost surely. Therefore, G_{μ}(x) → G_{λ}(x) for all x ∈ C_{G_{λ}}. Since G_{λ}(x) is continuous for all λ > 0 (see e. g. Lemma 2 in [11]), this means all x ∈ ℝ. Hence, for all x ∈ ℝ, \overline{G}_{n}(s, x) → G_{s}(x)/s for s ∈ ( 1/c, 1] and \overline{G}_{n}(s, x) → 0 for s ∈ ( 0, ∞) \ [ 1/c, 1 ]. Thus

∑_{r=k_{m−1}+1}^{k_{m}} \frac{G_{r/k_{m}}(x)}{r/k_{m}} \frac{1}{k_{m}} = ∫_{0}^{∞} \overline{G}_{m}(s, x) ds → ∫ 1_{\frac{1}{c}} \frac{G_{γ}(x)}{γ} dγ,

as m → ∞, by Lebesgue’s theorem and, since (1.4) is equivalent to n log c/ log k_{n} → 1, all this together yields (2.4) along {k_{n}}. But log k_{n+1}/ log k_{n} → 1 and the terms G_{γ_{n}}(x)/γ_{n} are non-negative and bounded, thus (2.4) and (2.3) hold along {n} = ℕ, as well. It also follows that the limiting distribution function is continuous.

The equivalence of (2.3) and (1.9) will be shown by Theorem 1 of Berkes and Dehling [6]. Therefore, we must check the conditions of that theorem.

For the normalizing constants A_{n} = n^{1/α}ℓ(1/n) in (1.9), by the slow variation of ℓ, there obviously exists a positive constant C for every δ ∈ (0, 1/α) such that

A_{l}/A_{k} ≥ C(l/k)^{δ}, l > k,

fulfilling the condition (2.2) in [6]. Condition (2.1) of [6] follows by Lemma 1, since (2.2) above is obviously implied by Theorem 1(i) in [11], and (2.1) can be seen by the slow variation of ℓ again.

The special case when the subsequence {n_{k}} is the whole sequence ℕ in the following lemma is exactly the statement of Lemma 2 in [5]. While the proof of Theorem 2 requires only this special case, the stronger subsequential variant here is needed to prove Theorem 5 in the next section.

Lemma 2. Let {X_{n}} be a sequence of independent identically distributed random variables with X_{n,n} = max_{1≤i≤n} X_{i}. Then for any subsequence {n_{k}} ⊂ ℕ and (proper or improper) distribution function Λ and any real sequences {a_{n}} and {b_{n}} the relations

lim_{k→∞} \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{1}{r} ℙ\{\frac{X_{r,r} − b_{r}}{a_{r}} ≤ x\} = Λ(x)  for any x ∈ C_{Λ}  (2.5)

and

lim_{k→∞} \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{1}{r} I\{\frac{X_{r,r} − b_{r}}{a_{r}} ≤ x\} = Λ(x)  a.s. for any x ∈ C_{Λ}  (2.6)

are equivalent.

Proof. Introduce M_{r}^{∗} := (X_{r,r} − b_{r})/a_{r}. It follows exactly the same way as in the proof of Theorem B in [4] (cf. also Lemma 2 in [5]) that for any bounded Lipschitz(1) function f on ℝ we have

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r}[f(M_{r}^{∗}) − 𝔼(f(M_{r}^{∗}))] → 0 a.s.  (2.7)

as n → ∞.

We assume first (2.5) and show (2.6). Fix some x ∈ C_{Λ} and choose a sequence ε_{m} → 0, ε_{m} > 0, such that x ± ε_{m} ∈ C_{Λ} for each m ∈ ℕ. Let the bounded Lipschitz(1) functions f_{m}^{x}(t) and g_{m}^{x}(t), t ∈ ℝ, be defined by

f_{m}^{x}(t) := \{ \begin{matrix}1, & if t ≤ x − ε_{m}, \\ 0, & if t ≥ x, \\ linear in between, \end{matrix}

and

g_{m}^{x}(t) := \{ \begin{matrix}1, & if t ≤ x, \\ 0, & if t ≥ x + ε_{m}, \\ linear in between. \end{matrix}

Clearly, f_{m}^{x}(t) ≤ I(t ≤ x) ≤ g_{m}^{x}(t) for all t ∈ ℝ, and hence

\frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{f_{m}^{x}(M_{r}^{∗}) − 𝔼(g_{m}^{x}(M_{r}^{∗}))}{r} ≤ θ_{k}(x) ≤ \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{g_{m}^{x}(M_{r}^{∗}) − 𝔼(f_{m}^{x}(M_{r}^{∗}))}{r} (2.8)

for each k, m ∈ ℕ, where

θ_{k}(x) = \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{I(M_{r}^{∗} ≤ x) − ℙ\{M_{r}^{∗} ≤ x\}}{r} .

Denoting by ξ_{k}^{m}(x) the lower bound in (2.8), we have

lim sup_{k→∞} ξ_{k}^{m}(x) = lim sup_{k→∞} \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{f_{m}^{x}(M_{r}^{∗}) − 𝔼(f_{m}^{x}(M_{r}^{∗}))}{r}

+ lim sup_{k→∞} \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{𝔼(f_{m}^{x}(M_{r}^{∗})) − 𝔼(g_{m}^{x}(M_{r}^{∗}))}{r}

≥ 0 + lim sup_{k→∞} \frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{ℙ\{M_{r}^{∗} ≤ x − ε_{m}\} − ℙ\{M_{r}^{∗} ≤ x + ε_{m}\}}{r}

= Λ(x − ε_{m}) − Λ(x + ε_{m}),

which goes to 0 as m → ∞. Here the inequality holds almost surely for each m ∈ ℕ by virtue of (2.7), and hence it holds on a set of probability 1 for all m ∈ ℕ. A similar estimate for the right-hand side of (2.8) shows that θ_{k}(x) → 0 a.s. as k → 0, that is, (2.6) follows from (2.5) as claimed. Assuming (2.6), the converse implication follows by taking expectation.

Proof of Theorem 2. First we show that

ϑ_{n}(x) := \frac{1}{log n} ∑_{r=1}^{n} \frac{H_{r}(x)}{r} → \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{K_{γ}(x)}{γ} dγ,  (2.9)

for all x ∈ ℝ. By the Merge Theorem in (1.11) and the definition of the Lévy distance, for every x ∈ ℝ and ε > 0 we have

lim inf_{n→∞} \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} K_{γ_{r}}(x − ε) − ε ≤ lim inf_{n→∞} ϑ_{n}(x)

≤ lim sup_{n→∞} ϑ_{n}(x) ≤ lim sup_{n→∞} \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} K_{γ_{r}}(x + ε) + ε,

whence (2.9) follows if we show that the limiting distribution function on the right-hand side of (2.9) is continuous and that

\frac{1}{log n} ∑_{r=1}^{n} \frac{K_{γ_{r}}(x)}{r} → \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{K_{γ}(x)}{γ} dγ

for all x ∈ ℝ. Although the distribution functions K_{γ}(·) are not continuous in general, the latter can be done along the same lines as in the proof of Theorem 1 above. Introduce \overline{K}_{m}(s, x) similarly as \overline{G}_{m}(s, x) in the proof of Theorem 1 but with K standing everywhere in place of G. Again, it is enough to infer that

∑_{r=k_{m−1}+1}^{k_{m}} \frac{K_{r/k_{m}}(x)}{r/k_{m}} \frac{1}{k_{m}} = ∫_{0}^{∞} \overline{K}_{m}(s, x) ds → ∫ 1_{\frac{1}{c}} \frac{K_{γ}(x)}{γ} dγ  (2.10)

as m → ∞. Using the definition of K_{γ}(·) in (1.10), it can be seen directly that K_{μ}(x) → K_{λ}(x) for each fixed x ∈ ℝ whenever μ ↑ λ if λ is a continuity point of K_{•}(x), or, equivalently by (1.10), if λ^{1/α}x is a continuity point of M_{R}. Since M_{R}(·) can have only a countable number of discontinuities, K_{μ}(x) → K_{λ}(x) holds for each x ∈ ℝ as μ ↑ λ for almost all λ > 0, that is, with an exceptional set of Lebesgue measure 0. Lebesgue’s dominated convergence theorem yields the desired convergence in (2.10) and the continuity of the limiting distribution function in (1.10) and (2.9) may be seen similarly. Now (1.12) follows from (2.9) by an application of Lemma 2.

3. General norming sequences: mixtures

In this section we are concerned with determining all possible limits if in (1.9) or (1.12) of Theorem 1 or 2, respectively, the norming factor r^{1/α}ℓ(1/r) is replaced by an arbitrary one, a_{r}, say. The motivation comes from [3], where it was shown that in the case when the random variables X_{1}, X_{2}, … have zero mean and finite variance then the class of all possible almost sure limiting distribution functions of the random functions

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{∑_{i=1}^{r} X_{i}}{a_{r}} ≤ x\} , x ∈ ℝ,

consists of all scale mixtures of the standard normal distribution function. Similar results were proved in [3], more generally, for the case when the underlying distribution is attracted to an arbitrary stable law.

Before stating the corresponding result for our semistable case here, we need to introduce some notation. Given c > 1, a non-negative function q(·) on the interval [1/c, 1] will be called a step-function if it is piecewise constant, i.e. there exist γ_{0} = 1/c < γ_{1} < ⋯ < γ_{k−1} < γ_{k} = 1 and non-negative numbers q_{1}, …, q_{k−1}, q_{k} such that q(γ) = ∑_{i=1}^{k} q_{i} I{γ ∈ [γ_{i−1}, γ_{i})} for some k ∈ ℕ. Let A be the set of such step-functions. For q ∈ A define

J(x) = J(q; x) = \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ} (q(γ)x)}{γ} dγ, x ∈ ℝ.

Define the class of functions J :={J(q; ·) : q ∈ A} and let J_{∗} be the set of finite mixtures of J ∈ J , i.e. the set of functions of the form

p_{1}J_{1}(·) + ⋯ + p_{N}J_{N}(·), where p_{1}, …, p_{N} ≥ 0, ∑_{i=1}^{N} p_{i} = 1, J_{1}(·), …, J_{N}(·) ∈ J

for some N ∈ ℕ. Finally, let J^{∗} be the weak closure of J_{∗}.

Now we state all of our scaling results, the proofs of which follow after the statement of Theorem 5.

Theorem 3. Assume the condition of Theorem 1. Then there exists a positive numerical sequence {a_{n}} such that

lim_{n→∞} \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{∑_{j=1}^{r} X_{j} − r∫ 1− \frac{1}{r}_{\frac{1}{r}} Q(u) du}{a_{r}} ≤ x\} = J(x),  (3.1)

almost surely for all x ∈ C_{J}, if and only if J ∈ J^{∗}.

Turning to maxima again, construct the class K^{∗} similarly to J^{∗}, starting from the limiting distributions in Theorem 2. Let K be the set of functions

K(x) = K(q; x) = \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{K_{γ} (q(γ)x)}{γ} dγ, x ∈ ℝ,

where q ∈ A, let K_{∗} be the set of finite convex mixtures of K ∈ K and, finally, let K^{∗} be the weak closure of the class K_{∗}. The result for maxima is then the following.

Theorem 4. Assume the conditions of Theorem 2. Then there exists a positive numerical sequence {a_{n}} such that

lim_{n→∞} \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{X_{r,r}}{a_{r}} ≤ x\} = K(x),  (3.2)

almost surely for all x ∈ C_{K}, if and only if K ∈ K^{∗}.

Theorem 3 in [3] says that the class of almost sure limiting distributions of logarithmic averages pertaining to sums of independent identically distributed random variables from the domain of attraction of a stable law cannot be enlarged by taking subsequential limits. A special case of Theorem 1 in [11] shows that independent variables from the domain of geometric partial attraction of a semistable distribution function G_{1} also possess a certain closure or maximum property: the non-degenerate partial asymptotic laws of their sums along arbitrary subsequences all belong to the ‘family’ around G_{1} , i.e. they are of the form δG_{λ} + d, where δ, λ > 0, d ∈ ℝ. Theorem 5 below, which is an analogue of Theorem 3 in [3], shows that in a sense the maximum property is preserved for almost sure distributional limits as well, and even for maxima.

Theorem 5. Assume the conditions of Theorem 1 (or Theorem 2, respectively). If the logarithmic averages on the left-hand side of (3.1) (or of (3.2)) converge to some limit J(x) (or K(x)) almost surely for each x ∈ C_{J} (or x ∈ C_{K}) along some subsequence {n_{k}}_{k=1}^{∞}, then necessarily J ∈ J^{∗} (or K ∈ K^{∗}). Furthermore, there exists a universal positive norming sequence {a_{n}}_{n=1}^{∞} for which the totality of subsequential almost sure limits of the corresponding logarithmic averages is identical with the whole class J^{∗} (or K^{∗}, respectively).

It is natural to ask all sorts of questions concerning the limiting classes J^{∗} and K^{∗}. Are there more compact descriptions of them? Are all the proper distribution functions in J^{∗} infinitely divisible? Does it contain normal distribution functions? Does it even contain infinitely divisible distribution functions? All these questions are open. It is easy to see that J^{∗} and K^{∗} contain improper distribution functions, and we conjecture that J^{∗} contains proper distribution functions that are not infinitely divisible and it probably does not contain normal distribution functions. However, we see that all distribution functions of the form

J(x) = \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ} (q(γ)x)}{γ} dγ, x ∈ ℝ,

are in J^{∗} for any q ∈ L_{1}^{+} = L_{1}^{+}[ 1/c, 1 ], the class of non-negative integrable functions on [ 1/c, 1 ]. Hence, all distribution functions of the form

J(x) = \frac{1}{log c} ∫_{L_{1}^{+}} ∫ 1_{\frac{1}{c}} \frac{G_{γ} (q(γ)x)}{γ} dγ μ(dq), x ∈ ℝ,  (3.3)

are also in J^{∗} for any probability measure μ on L_{1}^{+} . Analogous statements hold for K^{∗}. But it is unclear whether the whole class J^{∗} is contained if in (3.3) we allow arbitrary sub-probability measures μ on L_{1}^{+} , and attach the weight 1 − μ(L_{1}^{+}) to the point x = 0, or whether the class thus obtained is wider than J^{∗}.

Turning now to the proofs, we begin with a lemma that is needed for the proof of Theorems 3 and 5.

Lemma 3. Let X_{1}, X_{2}, … be independent random variables, S_{n} = X_{1} + ⋯ + X_{n}, and let {A_{n}} and {B_{n}} be numerical sequences, {A_{n}} being positive, such that

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{S_{r} − B_{r}}{A_{r}} ≤ x\} → G(x)  a.s. for all x ∈ C_{G},  (3.4)

where G is a distribution function continuous at the origin. Assume that

A_{l}/A_{k} ≥ C_{1}(l/k)^{β},  l > k,  (3.5)

and

𝔼(|\frac{S_{n} − B_{n}}{A_{n}} |^{p}) ≤ C_{2},  n = 1, 2, …,  (3.6)

for some positive constants β, p, C_{1} and C_{2} . Then for any positive norming sequence {a_{n}} and any subsequence {n_{k}} ⊂ ℕ and (proper or improper) distribution function H the relations

\frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{1}{r} I\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} → H(x)  a.s. for all x ∈ C_{H}, as k → ∞,  (3.7)

and

\frac{1}{log n_{k}} ∑_{r=1}^{n_{k}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} → H(x)  for all x ∈ C_{H}, as k → ∞,  (3.8)

are equivalent.

Note before the proof that I can of course be replaced by ℙ in (3.4) by Theorem 1 of Berkes and Dehling [6] if {n_{k}} = ℕ, where the continuity assumption on G is not needed. The lemma shows that if G is continuous at the origin, then we can replace I by ℙ even if the norming factor is changed arbitrarily, and this can be done even along subsequences. Note also that the slight continuity condition is satisfied in our applications since the limiting distribution function in (1.9), inheriting this property from the distribution functions G_{γ}(·), 1/c ≤ γ ≤ 1, is everywhere continuous.

Proof. First we prove that if {a_{n}} satisfies the additional condition

a_{n} ≥ KA_{n} , n = 1, 2, …, for some positive constant K  (3.9)

then for any bounded Lipschitz(1) function f on ℝ we have

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r}[f(\frac{S_{r} − B_{r}}{a_{r}} ) − 𝔼(f(\frac{S_{r} − B_{r}}{a_{r}} ))] → 0 a.s. as n → ∞.  (3.10)

Set

ξ_{r}^{(f)} = f({S_{r} − B_{r}}/a_{r}) − 𝔼(f({S_{r} − B_{r}}/a_{r})).

By a standard argument in a.s. central limit theory, (3.10) will follow if we show that

|𝔼(ξ_{k}^{(f)}ξ_{l}^{(f)})| ≤ C(\frac{k}{l} )^{ρ}, k ≤ l,

for some constants C > 0 and ρ > 0. Choosing the constant K_{1} so that |f(x)| ≤ K_{1} , |f(x) − f(y)| ≤ K_{1}|x − y| hold for all x, y, we get for any k ≤ l, using (3.5), (3.6) (note that in (3.6) we can assume p < 1) and the independence of S_{k} and S_{l} − S_{k},

|𝔼(ξ_{k}^{(f)}ξ_{l}^{(f)})| = |Cov(f(\frac{S_{k} − B_{k}}{a_{k}} ), f(\frac{S_{l} − B_{l}}{a_{l}} ))|

= |Cov(f(\frac{S_{k} − B_{k}}{a_{k}} ), f(\frac{S_{l} − B_{l}}{a_{l}} ) − f(\frac{(S_{l} − S_{k}) − (B_{l} − B_{k})}{a_{l}}))|

≤ 4K_{1} 𝔼(K_{1}|\frac{S_{k} − B_{k}}{a_{l}} | ∧ 2K_{1}) ≤ 8K_{1}^{2} 𝔼(|\frac{S_{k} − a_{k}}{KA_{l}} | ∧ 1)

≤ 8K_{1}^{2} 𝔼(|\frac{S_{k} − B_{k}}{KA_{l}} |^{p}) ≤ 8K_{1}^{2}C_{2}K^{−p}(\frac{A_{k}}{A_{l}} )^{p} ≤ C(\frac{k}{l} )^{βp} .

Dropping now condition (3.9), we first deal with the case {n_{k}} = ℕ; instead of n_{k} we simply write n. Let {a_{n}} be an arbitrary positive numerical sequence and assume that (3.7) holds. Fix a continuity point x > 0 of H (negative x’s can be handled similarly). Let ε > 0 and assume that x ± ε and ±εx are also continuity points of H (this excludes only countably many ε’s). Set

a_{n}^{∗} := \{ \begin{matrix}a_{n} & if a_{n} ≥ εA_{n}, \\ εA_{n} & if a_{n} < εA_{n}, \end{matrix}

and let f_{ε,x}(t) denote the bounded Lipschitz(1) function which is 1 for t ≤ x, 0 for t ≥ x + ε and linear in [x, x + ε]. We consider the six expressions

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} , \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{S_{r} − B_{r}}{a_{r}^{∗}} ≤ x\} ,

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} f_{ε,x}(\frac{S_{r} − B_{r}}{a_{r}^{∗}} ) , \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} 𝔼(f_{ε,x}(\frac{S_{r} − B_{r}}{a_{r}^{∗}} )),

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}^{∗}} ≤ x\} , \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} ,

(3.11)

and estimate the difference between the consecutive ones. Clearly, the difference

|I{(S_{r} − B_{r})/a_{r} ≤ x} − I{(S_{r} − B_{r})/a_{r}^{∗} ≤ x}|

equals I{a_{r}x/A_{r} < (S_{r} − B_{r})/A_{r} ≤ εx} ≤ I{|(S_{r} − B_{r})/A_{r}| ≤ εx} for a_{r} < εA_{r} and thus the difference of the first two expressions in (8) is at most

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{|\frac{S_{r} − B_{r}}{A_{r}} | ≤ εx\} .  (3.12)

By (3.4), and since ±εx are continuity points of G, the expression in (3.12) converges a.s. to ψ(ε) := G(εx) − G(−εx). Since G is continuous at the origin, ψ(ε) → 0 if ε → 0. Thus the limsup of the difference of the first two terms in (3.11) is at most ψ(ε). Replacing I by ℙ, the same estimate applies for the difference of the last two expressions of (3.11). Next we observe that

I_{(−∞,x]} ≤ f_{ε,x} ≤ I_{(−∞,x+ε]}  (3.13)

and thus the difference of the second and third expressions in (3.11) is bounded by

|\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{S_{r} − B_{r}}{a_{r}^{∗}} ≤ x\} − \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} I\{\frac{S_{r} − B_{r}}{a_{r}^{∗}} ≤ x + ε\}| .  (3.14)

Relation (3.7) and the estimate for the difference of the first two expressions in (3.11) imply that for almost every outcome of the sample space and for all sufficiently large n the first and second terms in (3.11) are within 2ψ(ε) of H(x) and H(x+ε), respectively, and thus the difference in (3.14) is ≤ |H(x+ε)−H(x)|+4ψ(ε) =: ψ_{1}(ε). Since a_{n}^{∗} ≥ εA_{n}, the statement proved at the beginning of the proof shows that the difference between the third and fourth term in (3.11) tends to 0 a.s. as n → ∞. Finally, the difference of the fourth and fifth expression in (3.11) can be estimated similarly as the difference of the second and the third, only instead of (3.13) we use the inequality f_{ε,x−ε} ≤ I_{(−∞,x]} ≤ f_{ε,x}.

Hence we proved that for sufficiently large n the difference of the first and last expressions in (3.11) is ≤ ψ_{2}(ε) where ψ_{2}(ε) → 0 as ε → 0. Thus (3.7) implies (3.8), at least if {n_{k}} = ℕ. But if {n_{k}} ≠ ℕ, then we only have to write n_{k} in place of n everywhere in (3.11) and apply the same estimates as above, and the implication (3.10)⇒(3.11) follows once again. The converse statement can be proved similarly.

Proof of Theorem 3. By Lemma 3 it suffices to prove that, under the condition of Theorem 1, there exists a positive numerical sequence {a_{n}} such that

lim_{n→∞} \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} ℙ\{\frac{∑_{j=1}^{r} X_{j} − r∫ 1− \frac{1}{r}_{\frac{1}{r}} Q(u) du}{a_{r}} ≤ x\} = J(x),  (3.15)

for all x ∈ C_{J}, if and only if J ∈ J^{∗}.

First assume that (3.15) holds for some {a_{n}} and J. We show that J ∈ J^{∗}. As in the proof of Theorem 1, let S_{n} = X_{1} + ⋯ + X_{n}, A_{n} = n^{1/α}ℓ(1/n), B_{n} = n∫ 1− \frac{1}{n}_{\frac{1}{n}} Q(u) du and put q_{n} = a_{n}/A_{n}. Let {k_{n}} be a sequence satisfying (1.4). Then by the Merge Theorem in (1.8),

\begin{aligned}∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} & = ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{A_{r}} ≤ \frac{xa_{r}}{A_{r}} \} \\ & = ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} G_{γ_{r}}(\frac{xa_{r}}{A_{r}} ) + o(1) \\ & = ∑_{r=k_{m−1}+1}^{k_{m}} \frac{G_{r/k_{m}}(q_{r}x)}{r} + o(1) \\ & = ∫_{k_{m−1}+1}^{k_{m}} \frac{G_{⌊t⌋/k_{m}}(q_{⌊t⌋}x)}{⌊t⌋} dt + o(1) \\ & = ∫_{k_{m−1}+1}^{k_{m}} \frac{G_{t/k_{m}}(q_{⌊t⌋}x)}{t} dt + o(1) \\ & = ∫ 1_{\frac{1}{c}} \frac{G_{γ}(q_{m}(γ)x)}{γ} dγ + o(1) , \end{aligned}  (3.16)

where q_{m}(γ) = q_{⌊k_{m}γ⌋}, with ⌊t⌋ standing for the integer part of t, and hence q_{m} ∈ A. Now put

J_{m}(x) = \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ}(q_{m}(γ)x)}{γ} dγ.

Then by (3.15), using also that log k_{n} ∼ n log c, we have

J(x) = lim_{N→∞} \frac{1}{log k_{N}} ∑_{m=1}^{N} ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\}

= lim_{N→∞} \frac{1}{N} ∑_{m=1}^{N} J_{m}(x),

i.e. J is the limit of convex linear combinations of J_{m} ∈ J , whence J ∈ J^{∗}. We prove the converse statement in several steps. First assume that J = J(q; x) ∈ J for some q ∈ A. Let {k_{m}} satisfy (1.4). We

can find a sequence {q_{m}(γ)} of step-functions such that the break-points of q_{m}(γ) are of the form ν/k_{m} with integer ν, q_{m}(γ) > 0 and q_{m}(γ) → q(γ) weakly (pointwise at each continuity point of q). If (k_{m−1} + 1)/k_{m} < c^{−1} , then we let q_{m}(γ) = q_{m}(1/c) for γ ∈ ((k_{m−1} + 1)/k_{m}, 1/c). Put

a_{r} = A_{r}q_{m}(\frac{r}{k_{m}} ), k_{m−1} + 1 ≤ r < k_{m} , m = 1, 2, ….

Then reading (3.16) backwards, we get

∫ 1_{\frac{1}{c}} \frac{G_{γ}(q_{m}(γ)x)}{γ} dγ = ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} + o(1),

hence (3.15) holds along the subsequence {k_{m}}. This implies, as in the proof of Theorem 1, that (3.15) holds along {n} = ℕ as well.

Now let J(·) be of the form

J(·) = p_{1}J_{1}(·) + ⋯ + p_{N}J_{N}(·)

with J_{i}(·) ∈ J and let p_{i} > 0, i = 1, …, N, be rational numbers such that ∑_{i=1}^{N} p_{i} = 1. By the previous arguments, we can find a_{r}^{(i)} such that

\frac{1}{log c} ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}^{(i)}} ≤ x\} = J_{i}(x) + o(1), x ∈ C_{J_{i}} .

In particular, suppose that p_{i} = β_{i}/β for some β_{i} ∈ ℕ, i = 1, …, N, with ∑_{i=1}^{N} β_{i} = β. Define a_{r} = a_{r}^{(i)} for all r and m satisfying k_{m−1} + 1 ≤ r ≤ k_{m} and m ≡ j (mod β), where j ∈ ℕ is such that ∑_{l=1}^{i−1} β_{l} ≤ j ≤ ∑_{l=1}^{i} β_{l} − 1, i = 1, …, N. Then, clearly,

lim_{n→∞} \frac{1}{βn log c} ∑_{m=1}^{βn} ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{S_{r} − B_{r}}{a_{r}} ≤ x\} = p_{1}J_{1}(x) + ⋯ + p_{N}J_{N}(x) = J(x)

for all x ∈ C_{J}, and so (3.15) holds along the subsequence {k_{βn}}. Thus it also holds along the whole sequence {n} = ℕ.

Finally, since the functions J ∈ J_{∗} with rational coefficients are weakly dense in J^{∗}, we can find a sequence {a_{n}} such that (3.15) holds. For details on this point, see [3]. This completes the proof of Theorem 3.

Proof of Theorem 4. We proceed similarly as above. Referring to Lemma 2, it is sufficient to show that under the stated conditions there exists a numerical sequence {a_{n}} such that

lim_{n→∞} \frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} ℙ\{\frac{X_{r,r}}{a_{r}} ≤ x\} = K(x)  (3.17)

for all x ∈ C_{K}, if and only if K ∈ K^{∗}. We first assume that (3.17) holds for some K and show that K ∈ K^{∗}. Using the Merge Theorem for Maxima at (1.11) we obtain similarly as in (3.16) that for any ε > 0 there exists a threshold number M = M(ε) and step-functions q_{m} such that for m ≥ M,

\begin{aligned}∫ 1_{\frac{1}{c}} \frac{K_{γ}(q_{m}(γ)(x − ε))}{γ} dγ − ε & ≤ ∑_{r=k_{m−1}+1}^{k_{m}} \frac{1}{r} ℙ\{\frac{X_{r,r}}{a_{r}} ≤ x\} \\ & ≤ ∫ 1_{\frac{1}{c}} \frac{K_{γ}(q_{m}(γ)(x + ε))}{γ} dγ + ε. \end{aligned}  (3.18)

Introduce

K_{m}(x) = \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{K_{γ}(q_{m}(γ)x)}{γ} dγ, x ∈ ℝ.

By definition, K_{m} ∈ K. Using (3.18),

lim sup_{N→∞} \frac{1}{N} ∑_{m=1}^{N} K_{m}(x − ε) − ε ≤ K(x) ≤ lim inf_{N→∞} \frac{1}{N} ∑_{m=1}^{N} K_{m}(x + ε) + ε,

which is to say that L(\frac{1}{N} ∑_{m=1}^{N} K_{m} , K) → 0 as N → ∞. Thus K ∈ K^{∗}, as stated.

The converse statement follows by a straightforward modification of the construction in the proof of the preceding theorem.

Proof of Theorem 5. The statements concerning the subsequential maximum property of the classes J^{∗} and K^{∗} are implicit in the proofs of Theorems 3 and 4, respectively: use the literally subsequential original forms of Lemmas 3 and 2, and then see (3.16) or (3.18), respectively. Now, let J_{†} consist of distribution functions of the form

J(x) = p_{1}J(q_{1} ; x) + ⋯ + p_{N}J(q_{N} ; x), x ∈ ℝ,

where N ∈ ℕ, p_{i} > 0, p_{i} ∈ ℚ, i = 1, …, N, ∑_{i=1}^{N} p_{i} = 1, and

J(q_{i}; x) = \frac{1}{log c} ∫ 1_{\frac{1}{c}} \frac{G_{γ}(q_{i}(γ)x)}{γ} dγ,

where q_{i} ∈ A is a non-negative rational-valued step-function with rational jump-points, i = 1, …, N. Thus J_{†} is a “rational restriction” of J_{∗}. Clearly, J_{†} is countable and its weak closure is J^{∗}. Hence it suffices to construct a norming sequence {a_{n}} for which the set of subsequential limits of the averages

\frac{1}{log n} ∑_{r=1}^{n} \frac{1}{r} ℙ\{\frac{∑_{j=1}^{r} X_{j} − r∫ 1− \frac{1}{r}_{\frac{1}{r}} Q(u) du}{a_{r}} ≤ x\}

contains J_{†}. The construction of such a sequence {a_{n}} can be done by a straightforward modification of the corresponding part in the proof of Theorem 3 in [3]. This completes the proof for sample sums. The case of sample maxima is entirely similar.

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