Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Proof. Since k>c(ε) implies  , it follows that

Lemma 2.4. Letθ>0, and let h be a positive, real valued function, such that . For anyC>0,

Proof. The result follows by the dominated convergence, since

The following lemma follows immediately from Spătaru [28, Lemma 2].
Lemma 2.5. For 1<β⩽2 andx,y>0,

Next, some purely computational auxiliary results.
Lemma 2.6. Forδ⩾−d+1 we have

Proof. We consider only the case δ⩾0; the case −d+1⩽δ<0 being similar is left to the reader. For k>2 we have
equation(2.5)

and for 2∨eδ⩽j<k we have
equation(2.6)

Combining (2.5) and (2.6), we obtain
equation(2.7)

Since   as k→∞ for a>−1, we have

The right-hand inequality of (2.7) provides the same asymptotics, and we are done. □
Lemma 2.7. Forγ>−1 we have, ask→∞,

Proof. For k>1 we have  . Next we observe that   lies between M(k)kγ−γ∑i₌₁k⁻¹M(i)iγ⁻¹ and M(k)kγ−γ∑i₌₁k⁻¹M(i)(i+1)γ⁻¹, since jγ−kγlies between   and −γ∑i₌jk⁻¹(i+1)γ⁻¹. In view of the fact that   as k→∞, we therefore obtain

The bound   provides the same asymptotics as k→∞. □

3. Proof of Theorem 1

Recall that G is the distribution function of a nondegenerate stable distribution with characteristic exponent α, 1<α⩽2, and Ψ(x)=1−G(x)+G(−x), x⩾0.

3.1. F=G

Lemma 3.1. Ford⩾2 andC>0,

Proof. Partial integration and Lemma 2.4 with h(y)=−Ψ′(y), yield

Proposition 3.1. For 1⩽p<α⩽2,

Proof. Let 0<δ<1. By making use of Lemma 2.6, choose k₀ such that
equation(3.1)

Moreover, assume that the function   is decreasing for x⩾k₀−1. Then, as G is also the distribution function of  ,
which, together with Lemma 3.1, yields
equation(3.2)

Analogously, we have
equation(3.3)

Since δ is at our disposal, (3.2) and (3.3) together finish the proof. □

3.2. F is attracted to a stable distribution

In this case bn takes the form bn=n^1/αhn, where {hn} is slowly varying in the sense of Karamata. Let β and b(ε) be as in Lemma 2.1. Also recall that Ψ^∗(x)=1−G^∗(x)+G^∗(−x), x⩾0.
Proposition 3.2.

Proof. The conclusion follows from the fact that

which, in view of Lemma 2.6 with δ=0, shows that

and, hence, that

Proposition 3.3.

Proof. Noting that Ψ^∗(x)⩽Cx^−α for x>0, that bj⩽Cj^1/α⁺γ^/2 for any γ>0 due to the slow variation of {hn}, it follows that

Next, in view of (2.2), we have d(j)⩽Cj^αγ/2 for any γ>0, and so, by choosing γ<(α−β)/αβ, the conclusion follows via

Proposition 3.4.

Proof. Specializing Lemma 2.5 at x=εj^1/p and y=εj^1/p/β yields

An application of Lemma 2.1 to the first term in the right-hand side above, and the fact that
equation(3.4)

for η>0 and k⩾1, applied to the second one, shows that

The conclusion follows. □

4. Proof of Theorem 2

As in the previous section, G is the distribution function of a nondegenerate stable distribution with characteristic exponent α, 1<α⩽2. Moreover, Z is a random variable with this distribution, that is, we can writeΨ(x)=P(|Z|>x), x⩾0.

4.1. F=G

Lemma 4.1. Ford⩾2 and 0<γ<α,

Proof. Immediate from Lemma 2.4 with h(y)=yγ⁻¹Ψ(y). □
Proposition 4.1. For 1⩽p<r<α,

Proof. Let 0<δ<1. By making use of Lemma 2.7 with γ=r/p−2, choose k₀ such that

and also so that the function   is decreasing for x⩾k₀−1. Following the path of the proof of Proposition 3.1 we now obtain

An application of Lemma 4.1 with   now yields

which together with an analogous lower bound for the   (cf. (3.3)) and the arbitrariness of δcompletes the proof. □

4.2. F is in the normal domain of attraction to a stable law

Now the normalizing constants are bj=Cj^1/α, j⩾1. Put a(ε)=ε⁻αp^/(α⁻p⁾.
Proposition 4.2.

Proof. Let M be a positive number, and set   asj→∞. Following the proof of Proposition 3.2, with jr^/p⁻² replacing 1/j and a(ε)M replacing b(ε), we first conclude, via Lemma 2.7 with γ=r/p−2, that

Letting ε↘0, we then obtain
equation(4.1)

Next, we observe that the moments of order <α of the normalized partial sums are uniformly bounded by Lemma 5.2.2 in [16, p. 142], which, together with Markov's inequality (see also [26, p. 163]), shows that, for all x≠0 and η<α,

With η=r and bj=Cj^1/α, we therefore conclude, via (3.4), that

and, hence, that
equation(4.2)

Finally, (4.1) and (4.2) together yield the desired conclusion. □

5. Proof of Theorem 3

5.1. F is normal

We thus assume w.l.o.g. that σ²=1. Also, N is a standard normal random variable, F its distribution functionΦ, and Ψ(x)=1−Φ(x)+Φ(−x)=P(|N|>x), x⩾0, in this subsection.
Lemma 5.1. Ford⩾2 andγ>0,

Proof. Immediate from Lemma 2.4 with h(y)=yγ⁻¹Ψ(y). □
Proposition 5.1. Forr⩾2 and 1⩽p<2,

Proof. For r/p<2 the conclusion follows as in Proposition 4.1 with α replaced by 2 and Z by N. Therefore, letr⩾2p. By Lemma 2.7 with γ=r/p−2, choose k₀ such that, for 0<δ<1,

Moreover, assume that   forj⩾k₀. Following the proof of Proposition 4.1 we now obtain

which, in view of Lemma 5.1 with  , yields

The conclusion follows as above. □

5.2. The general case

We thus consider i.i.d. random variables with mean 0, variance 1, under the moment assumption thatE[|X|r(log(1+|X|))d⁻¹]<∞. Also, recall that ρ(ε)=ε⁻²p^/(2−p⁾. The proof of the next proposition follows closely the pattern of the proof of the first part of Proposition 4.2, and is therefore omitted.
Proposition 5.2.

Proposition 5.3.

Proof. This is a special case of the next result. □
Proposition 5.4.

Proof. Let M>1. Lemma 2.5 with x=εj^1/p, y=εj^1/p/γ with γ=r/(2−p), and β=2, together with an application ofLemma 2.2 and (3.4), yields

The conclusion follows, in view of the fact that  . □

6. Proof of Theorem 4

6.1. F is normal

We use the notation and assumptions from Section 5.1. Also, 0⩽δ⩽1.
Proposition 6.1.

Proof. Let 0<η<1. By making use of Lemma 2.6, choose k₀ such that

and also such that   is decreasing for x⩾k₀−1. Then

This establishes the upper bound for the  . The lower bound for the   follows as before. □

6.2. The general case

We thus consider i.i.d. random variables with mean 0 and variance 1. Also, recall that c(ε)=eM^/ε², where M>1.
Proposition 6.2.

Proof. Let   as j→∞. Following the pattern of the previous proofs, it follows, via Lemma 2.6, that

Proposition 6.3.

Proof. Immediate from the next result. □
Proposition 6.4.

Proof. Lemma 2.5 with  ,  , and β=2, Lemma 2.3 and the fact that

yield

7. Proof of Theorem 5

7.1. F is normal

We thus assume w.l.o.g. that σ²=1. Also, F is the standard normal distribution function Φ, andΨ(x)=1−Φ(x)+Φ(−x), x⩾0 in this subsection.
Proposition 7.1.

Proof. Let 0<η<1. By making use of Lemma 2.6 with δ=−1, choose k₀⩾3 such that

Moreover, assume that   is decreasing for x⩾k₀−1, and set  . Then

via partial integration. The proof is concluded the usual way. □

7.2. The general case

Thus X,X₁,X₂,… are i.i.d. random variables with mean 0 and variance 1.
Proposition 7.2. Suppose that for someδ>1.Then

Proof. Choose A so large that   is increasing for x⩾A, and define

By Theorem 5.6 in [26, p. 151], for j⩾A², we then have

which, since

shows that

8. Some further results and remarks

8.1. Some corollaries

Even though it is not true that   as j→∞, a substantial part of the proofs is devoted to “replacing” d(j) by   in Lemma 2.1, Lemma 2.2 and Lemma 2.3, and by   in Propositions 1 of 3, 4, 5, 6 and 7. With this in mind, an investigation of the proofs shows that, by replacing d(j) by (logj)θ in Lemma 2.1, Lemma 2.2 and Lemma 2.3, where θ is a positive real number, we obtain variations of those lemmas which therefore are much easier to prove. As an illustration we state one of them; the analog of Lemma 2.1.
Lemma 8.1. Suppose thatE[|X|β(log(1+|X|))d⁻¹<∞ and setb(ε)=ε⁻βp^/(β⁻p⁾, where 1⩽p<β<α. For any constanta>0,

Secondly, by replacingd(j) by  in the propositions (thus, without the factorial ), we obtain analogous modifications, which, again, are more easily established.
Thus, let X and   be i.i.d. random variables with mean 0 and partial sums  . The following corollaries emerge.
Corollary 1. Let 1⩽p<α⩽2. Suppose that F belongs to the domain of attraction of a nondegenerate stable distribution G with characteristic exponentα. Then

Corollary 2. Let 1⩽p<r<α⩽2. Suppose that F belongs to the normal domain of attraction of a nondegenerate stable distribution G with characteristic exponentα, and let Z have distribution G. Then

Corollary 3. Suppose that , r⩾2, setEX²=σ², and let N denote a standard normal random variable. For 1⩽p<2,

Corollary 4. Suppose thatE[X²(log(1+|X|))θ]<∞, setEX²=σ², and let N denote a standard normal random variable. For 0⩽δ⩽1,

Corollary 5. Suppose that for someδ>1, and setEX²=σ². Then