## LARGE DEVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILED DEPENDENT RANDOM VECTORS*

### Adam Jakubowski Alexander V. Nagaev Alexander Zaigraev

### Nicholas Copernicus University

### Faculty of Mathematics and Computer Science ul. Chopina 12/18, 87-100 Toru´n, Poland

### E-mail: adjakubo@– nagaev@– alzaig@mat.uni.torun.pl

### ABSTRACT

### Necessary and sufficient conditions are given for multidimensional *p - stable limit theorems (i.e. theorems on convergence of normal-* *ized partial sums S* _{n} */b* _{n} of a stationary sequence of random vec- *tors to a non-degenerate strictly p-stable limiting law µ, with 1/p-* *regularly varying normalizing sequence b* *n* ). It is proved that sim- *ilarly as in the one-dimensional case the conditions for 0 < p < 2* consist of two parts: one responsible for (very weak) mixing prop- erties and another, describing asymptotics of probabilities of large *deviations (with a minor additional condition for p = 1). The pa-* per focuses on effective methods of proving such large deviation results.

_{n}

_{n}

### Key Words and Phrases: multivariate stable distributions, regular variation, stable limit theorems, large deviations, station- *ary sequences, ψ-mixing, φ-mixing, m-dependence.*

*∗*

### Supported by Komitet Bada´n Naukowych under Grant PB 591/P03/95/08

### 1. A MULTIDIMENSIONAL STABLE LIMIT THEO- REM

*Let X* _{1} *, X* _{2} *, . . . be a stationary sequence of d-dimensional ran-* *dom vectors with partial sums S* _{0} *= 0, S* _{n} = ^{Pn} _{j=1} *X* _{j} . Following *Jakubowski (1993) we will say that a p-stable limit theorem holds* *for {X* *j* *} if there exist a non-degenerate strictly p-stable law µ on* *IR* ^{d} *and a 1/p-regularly varying sequence b* *n* , such that

_{n}

^{Pn}

_{j=1}

_{j}

^{d}

*S* *n*

*b* _{n} *−→*

_{n}

*D* *µ,* *as n → +∞.* (1)

*Recall that a p-stable law µ is strictly stable if for all a, b > 0* *one can find c = c(a, b) > 0 such that (µ◦R* ^{−1} _{a} *)∗(µ◦R* ^{−1} _{b} *) = µ◦R* ^{−1} _{c} , *where for a > 0, R* _{a} *(x) = a · x is a rescaling of IR* ^{d} . For the case *p 6= 1, 2, the logarithm log ˆ* *µ(y) of the characteristic function of a* *strictly p-stable law can be written in the form*

^{−1}

_{a}

^{−1}

_{b}

^{−1}

_{c}

_{a}

^{d}

*η* _{p}

_{p}

### Z

*{s∈S*

^{d−1}*;hy,si>0}* *|hy, si|* ^{p} *κ(ds) + η* _{p}

^{p}

_{p}

### Z

*{s∈S*

^{d−1}*;hy,si<0}* *|hy, si|* ^{p} *κ(ds),* (2) *where y ∈ IR* ^{d} *, S* ^{d−1} *is the d − 1-dimensional unit sphere in IR* ^{d} , *κ is a finite Borel measure on S* ^{d−1} and

^{p}

^{d}

^{d−1}

^{d}

^{d−1}

*η* _{p} =

_{p}

###

###

###

###

###

### *Z ∞*

### 0 *(e* ^{iu} *− 1)u* ^{−(1+p)} *du* *if 0 < p < 1,*

^{iu}

^{−(1+p)}

*Z ∞*

### 0 *(e* ^{iu} *− 1 − iu)u* ^{−(1+p)} *du if 1 < p < 2.*

^{iu}

^{−(1+p)}

### (3)

*For p = 1, a law µ is strictly stable if it is a shift of a symmetric* strictly stable law with logarithm of the characteristic function of the form

*−(1/2)π* ^{Z}

*S*

^{d−1}*|hy, si|κ(ds),* (4)

*where κ is a symmetric measure on S* ^{d−1} . Let us denote by

^{d−1}

*Stab (p, κ) the strictly stable law with logarithm of the charac-*

### teristic function given by formulas (2) or (4). For more informa-

### tion on stable laws and processes we refer to Samorodnitsky and

### Taqqu (1994).

*For random variables (d = 1) Jakubowski (1993, 1997) ob-* *tained necessary and sufficient conditions for a p-stable limit the-* *orem to hold. In the case of heavy-tailed random variables (i.e. if* *0 < p < 2) the conditions essentially consist of two parts: a part* responsible for “mixing” properties (Condition B _{1} below) and a part describing asymptotic behaviour of probabilities of large de- viations (Condition LD 1 below). In both cases subscript 1 stands *for the dimension d = 1. Formal statements are as follows.*

*• Condition B* _{1} *. For each λ ∈ IR* ^{1} *, and as n → +∞*

*1≤k,l≤n* max

*k+l≤n*

*|E e* ^{iλ(S}

^{iλ(S}

^{k+l}^{/b}

^{/b}

^{n}^{)} *− Ee* ^{iλ(S}

^{iλ(S}

^{k}^{/b}

^{/b}

^{n}^{)} *· Ee* ^{iλ(S}

^{iλ(S}

^{l}^{/b}

^{/b}

^{n}^{)} *| −→ 0.* (5)

*• Condition LD* _{1} *. There exists a sequence r* _{n} *→ +∞ such* *that for all sequences x* _{n} *increasing to +∞ slowly enough* *(i.e. x* _{n} *= o(r* _{n} ))

_{n}

_{n}

_{n}

_{n}

*x* ^{p} _{n} *P (S* _{n} */b* _{n} *> x* _{n} *) −→ c* _{+} *,* *x* ^{p} _{n} *P (S* _{n} */b* _{n} *< −x* _{n} *) −→ c* _{−} *, (6)* *where 0 < c* + *+ c* *−* *< +∞ and b* *n* *→ +∞.*

^{p}

_{n}

_{n}

_{n}

_{n}

^{p}

_{n}

_{n}

_{n}

_{n}

_{−}

### The relations between Conditions B _{1} and LD _{1} *and p-stable* *limit theorems are particularly appealing in the case p 6= 1, as* the following theorem shows (see Jakubowski, 1993 for the case *0 < p < 1 and 1997 for the case 1 < p < 2).*

*Theorem 1 Let 0 < p < 1 or 1 < p < 2. Suppose Conditions B* 1

*and LD* _{1} *hold with b* _{n} *→ +∞ and 0 < c* _{+} *+ c* _{−} *< +∞. Then b* _{n} *varies 1/p-regularly and as n → +∞*

_{n}

_{−}

_{n}

*S* _{n}

_{n}

*b* _{n} *−→*

_{n}

*D* *Stab (p, κ* _{(c}

_{(c}

_{+}

_{,c}

_{,c}

_{−}_{)} *),* (7) *where κ* _{(c}

_{(c}

_{+}

_{,c}

_{,c}

_{−}_{)} *{+1} = c* _{+} *and κ* _{(c}

_{(c}

_{+}

_{,c}

_{,c}

_{−}_{)} *{−1} = c* _{−} *.*

_{−}

*Conversely, (7) with c* _{+} *+ c* _{−} *> 0 and 1/p-regular variation of*

_{−}

*b* _{n} *imply Conditions B* _{1} *and LD* _{1} *.*

_{n}

*Notice that for d = 1 we have S* ^{d−1} *= S* ^{0} *= {−1, +1}.*

^{d−1}

*For p = 1 a minor additional assumption on centering is neces-* sary (see Theorem 2.2, Jakubowski, 1997), which is automatically *satisfied, when the S* _{n} ’s are symmetric:

_{n}

*Theorem 2 If p = 1 and for each n ∈ IN, the law of S* _{n} *is* *symmetric, then Condition B* _{1} *and Condition LD* _{1} *with 0 < c* _{+} = *c* _{−} *= c < +∞ hold if, and only if,*

_{n}

_{−}

*S* _{n}

_{n}

*b* _{n} *−→*

_{n}

*D* *Stab (1, κ* _{(c,c)} *),* *as n → +∞,* (8) *where 0 < c < +∞ and b* _{n} *is regularly varying with exponent 1.*

_{(c,c)}

_{n}

### The purpose of the present note is to prove a multidimensional generalization of the above results.

### Let us begin with introducing a multidimensional version of Condition B _{1} .

*• Condition B* *d* *. For each y ∈ IR* ^{d} *, and as n → +∞*

^{d}

### max

*1≤k,l≤n* *k+l≤n*

*|E e* ^{ihy,S}

^{ihy,S}

^{k+l}^{/b}

^{/b}

^{n}^{i} *− Ee* ^{ihy,S}

^{i}

^{ihy,S}

^{k}^{/b}

^{/b}

^{n}^{i} *· Ee* ^{ihy,S}

^{i}

^{ihy,S}

^{l}^{/b}

^{/b}

^{n}^{i} *| −→ 0.* (9)

^{i}

*Condition B* _{d} describes a kind of “asymptotic independence”

_{d}

### of partial sums. It is however essentially weaker than mixing *conditions (such as α-mixing) usually considered in limit theory* for sums, for there exist non-ergodic sequences satisfying (9). On the other hand Condition B _{d} *held for each y ∈ IR* ^{d} implies (under *mild additional assumptions) uniform convergence over bounded* *subsets of IR* ^{d} :

_{d}

^{d}

^{d}

*1≤k,l≤n* max

*k+l≤n*

### sup

*kyk≤K* *|E e* ^{ihy,S}

^{ihy,S}

^{k+l}^{/b}

^{/b}

^{n}^{i} *− Ee* ^{ihy,S}

^{i}

^{ihy,S}

^{k}^{/b}

^{/b}

^{n}^{i} *· Ee* ^{ihy,S}

^{i}

^{ihy,S}

^{l}^{/b}

^{/b}

^{n}^{i} *| −→ 0, (10)*

^{i}

*for every K > 0 and as n → +∞. In particular, (10) implies*

### that given Condition B _{d} *for some normalizing sequence {b* _{n} *}, we*

_{d}

_{n}

*obtain it for all sequences b* ^{0} _{n} *such that b* _{n} *≤ Cb* ^{0} _{n} *, n ∈ IN, for some*

^{0}

_{n}

_{n}

^{0}

_{n}

*constant C > 0. This has been observed by Szewczak (1996).*

### For examples of sequences satisfying Condition B _{d} and further *discussion in the case d = 1 (which can be easily extended to* several dimensions) we refer to Jakubowski (1991,1993).

_{d}

### The form of Condition LD _{d} is somewhat more complicated than (6) and involves convergence to a measure which is, in gen- *eral, finite only outside of every neighborhood of 0 ∈ IR* ^{d} (hence *σ-finite). In our theorems such measures will always be L´evy* measures, but from the point of view of sufficiency of Condition LD _{d} it is reasonable to formulate this condition in full generality.

_{d}

^{d}

_{d}

*For further purposes, let us denote by ν(p, κ) the L´evy mea-* *sure of the infinite divisible law Stab (p, κ). This means that for*

*“radial” sets A of the form A = ∪* _{x∈B} *x · V , where B ∈ B* _{IR}

_{x∈B}

_{IR}

^{+}

### and *V ∈ B* _{S}

_{S}

^{d−1}### , we have

*ν(p, κ)(A) =* ^{Z}

*B* *u* ^{−1−p} *du · κ(V ).* (11) *Clearly, ν(p, κ) = 0 if, and only if, κ = 0 and ν(p, κ) is symmetric* *if, and only if, κ is symmetric.*

^{−1−p}

*• Condition LD* _{d} *. There exists a sequence b* _{n} *→ +∞ and a* *measure ν on IR* ^{d} *, finite outside of every neighborhood of 0 ∈* *IR* ^{d} *, such that for all sequences x* *n* *→ +∞ increasing “slowly* *enough” (i.e. x* _{n} *= o(r* _{n} *) for some sequence r* _{n} *→ +∞) we* have

_{d}

_{n}

^{d}

^{d}

_{n}

_{n}

_{n}

*x* ^{p} _{n} *P (S* _{n} */b* _{n} *∈ x* _{n} *A) −→ ν(A),* (12) *whenever A ∈ B* ^{d} *, A 63 0 and ν(∂A) = 0.*

^{p}

_{n}

_{n}

_{n}

_{n}

^{d}

### Given Conditions B _{d} and LD _{d} we have a complete generaliza- tion of Theorems 1 and 2.

_{d}

_{d}

*Theorem 3 Let 0 < p < 1 or 1 < p < 2. Suppose Conditions B* _{d} *and LD* _{d} *hold with b* _{n} *→ +∞ and ν 6= 0.*

_{d}

_{d}

_{n}

*Then b* _{n} *varies 1/p-regularly, ν = ν(p, κ) for some κ 6= 0 and* *S* _{n}

_{n}

_{n}

*b* _{n} *−→*

_{n}

*D* *Stab (p, κ),* *as n → +∞.* (13)

*Conversely, (13) with κ 6= 0 and 1/p-regular variation of b* _{n} *imply Conditions B* _{d} *and LD* _{d} *with ν = ν(p, κ).*

_{n}

_{d}

_{d}

*Theorem 4 Let p = 1. Suppose for each n ∈ IN, the law of* *S* _{n} *is symmetric. Then Condition B* _{d} *and Condition LD* _{d} *with* *symmetric ν = ν(1, κ) 6= 0 hold if, and only if,*

_{n}

_{d}

_{d}

*S* _{n}

_{n}

*b* _{n} *−→*

_{n}

*D* *Stab (1, κ),* *as n → +∞,* (14) *where κ 6= 0 is symmetric and b* _{n} *is regularly varying with expo-* *nent 1.*

_{n}

### Proof. Necessity of Condition LD _{d} . In order to prove Con- dition LD _{d} we shall proceed similarly as in the one-dimensional case.

_{d}

_{d}

*Let for each n, Y* *n,1* *, Y* *n,2* *, . . . be independent copies of S* *n* */b* *n* . *By strict stability of µ,*

*k* ^{−1/p} ^{X} ^{k}

^{−1/p}

^{k}

*j=1*

*Y* _{n,j} *−→*

_{n,j}

*D* *µ,* *as n → +∞,* *k = 1, 2, . . . .* (15) *It follows that there exists r* _{n} *% +∞ such, that for every sequence* *{k* _{n} *} ⊂ IN, which is increasing to infinity slowly enough, i.e.,* *k* _{n} *→ +∞, k* _{n} *= o(r* _{n} ), we have

_{n}

_{n}

_{n}

_{n}

_{n}

*k* ^{−1/p} _{n} ^{X} ^{k}

^{−1/p}

_{n}

^{k}

^{n}*j=1*

*Y* _{n,j} *−→*

_{n,j}

*D* *µ,* *as n → +∞.* (16)

### (Notice that condition (16) is considerably weaker than condition (15)).

*Since k* _{n} *→ ∞, the array {k* _{n} ^{−1/p} *Y* _{n,j} *} of row-wise independent* random variables is infinitesimal and we can apply a convergence criterion for stable laws for sums of independent random vari- *ables. In particular, for each Borel subset A ∈ B* ^{d} which is sepa- *rated from zero and such that ν(∂A) = 0, we have as n → +∞*

_{n}

_{n}

^{−1/p}

_{n,j}

^{d}

*k* _{n} *P (S* _{n} */(b* _{n} *k* _{n} ^{1/p} *) ∈ A) = k* _{n} *P (S* _{n} */b* _{n} *∈ k* ^{1/p} _{n} *· A) −→ ν(A), (17)*

_{n}

_{n}

_{n}

_{n}

^{1/p}

_{n}

_{n}

_{n}

^{1/p}

_{n}

*where ν = ν(p, κ) is the L´evy measure of the stable law µ.*

*Setting x* _{n} *= k* _{n} ^{1/p} we obtain Condition LD _{d} with sequences *x* _{n} *of specific form and with rate r* ^{1/p} _{n} . Due to the special form *of the L´evy measure ν(p, κ) we can extend (17) to all sequences* *x* _{n} *→ +∞, x* _{n} *= o(r* ^{1/p} _{n} ).

_{n}

_{n}

^{1/p}

_{d}

_{n}

^{1/p}

_{n}

_{n}

_{n}

^{1/p}

_{n}

### Necessity of Condition B *d* *. Let y ∈ IR* ^{d} . Then *hy, S* _{n} */b* _{n} *i −→*

^{d}

_{n}

_{n}

*D* *µ* _{y} *,*

_{y}

*where µ* _{y} *is the strictly stable law on IR* ^{d} *being an image of µ* *under the mapping IR* ^{d} *3 x 7→ hy, xi ∈ IR* ^{1} *. If y ∈ IR* ^{d} is such *that µ* _{y} *is different from δ* _{0} , we obtain (9) from the corresponding *theorem for d = 1. If µ* _{y} *= δ* _{0} *, we have for any sequence k* _{n} *≤ n*

_{y}

^{d}

^{d}

^{d}

_{y}

_{y}

_{n}

### *

*y,* *S* *k*

*n*

*b* _{n}

_{n}

### +

### = *b* *k*

*n*

*b* _{n} *·*

_{n}

### *

*y,* *S* *k*

*n*

*b* _{k}

_{k}

_{n}### +

*−→* *P* *0,*

*for if k* _{n}

_{n}

^{0}*→ ∞ along some subsequence n* ^{0} , then sup _{n} *b* _{k}

^{0}

_{n}

_{k}

_{n}*/b* _{n} *< +∞*

_{n}

*by regular variation of b* _{n} *and if k* _{n}

_{n}

_{n}

^{00}### remains bounded along some *subsequence n* ^{00} *, then we have b* _{k}

^{00}

_{k}

_{n00}*/b* _{n}

_{n}

^{00}*→ 0 by b* _{n} *→ ∞. Hence* *if k* _{n} *+ l* _{n} *≤ n, then S* _{k}

_{n}

_{n}

_{n}

_{k}

_{n}_{+l}

_{+l}

_{n}*/b* _{n} *−→* _{P} *0, S* _{k}

_{n}

_{P}

_{k}

_{n}*/b* _{n} *−→* _{P} 0 and *S* _{l}

_{n}

_{P}

_{l}

_{n}*/b* _{n} *−→* _{P} *0 and (9) is satisfied for y, too.*

_{n}

_{P}

### Sufficiency. Suppose Conditions B _{d} and LD _{d} hold for some *b* *n* *→ ∞ and some measure ν which is finite outside of every* *neighborhood of 0 ∈ IR* ^{d} *. Since strict stability of µ is equivalent* *to µ* ^{∗n} *= µ ◦ R* ^{−1} _{n}

_{d}

_{d}

^{d}

^{∗n}

^{−1}

_{n}

*1/p*

*for each n ∈ IN, it is sufficient to prove that* *for each y ∈ IR* ^{d} one dimensional sums ^{Pn} _{k=1} *hy, X* _{k} */b* _{n} *i converge* *to some strictly p-stable law on IR* ^{1} (possibly degenerated at 0).

^{d}

^{Pn}

_{k=1}

_{k}

_{n}

*Let us fix y ∈ IR* ^{d} *, y 6= 0, and consider in (12) the following* *sets A* ^{y} _{+} *, A* ^{y} _{−} *⊂ IR* ^{d}

^{d}

^{y}

^{y}

_{−}

^{d}

*A* ^{y} _{+} *= {x; hy, xi > 1},* *A* ^{y} _{−} *= {x; hy, xi < −1}.* (18)

^{y}

^{y}

_{−}

### These sets are separated from zero and we may assume that

*ν(∂A* ^{y} _{±} *) = 0 (otherwise we may replace y with r · y for some*

^{y}

_{±}

*1 > r > 0). Moreover, by (12) we have* *x* ^{p} _{n} *P*

^{p}

_{n}

###

### X *n* *k=1*

*hy, X* _{k} */b* _{n} *i > x* _{n}

_{k}

_{n}

_{n}

###

### *= x* ^{p} _{n} *P (S* _{n} */b* _{n} *∈ x* _{n} *A* ^{y} _{+} *) → c* _{+} *= ν(A* ^{y} _{+} *),* *provided x* _{n} *→ +∞ slowly enough. Similar relation holds for the* left-hand tails of ^{Pn} _{k=1} *hy, X* _{k} */b* _{n} *i. It follows that in the case*

^{p}

_{n}

_{n}

_{n}

_{n}

^{y}

^{y}

_{n}

^{Pn}

_{k=1}

_{k}

_{n}

*ν(A* ^{y} _{+} *) + ν(A* ^{y} _{−} *) > 0* (19) *we may apply either Theorem 1 (for 0 < p < 1 and 1 < p < 2)* *or Theorem 2 (for p = 1) in order to get the convergence of* *{* ^{Pn} _{k=1} *hy, X* _{k} */b* _{n} *i} to a non-degenerate strictly p-stable law. In* *particular, b* *n* *is p-regularly varying for there are y’s satisfying* *(19) (by ν 6= 0).*

^{y}

^{y}

_{−}

^{Pn}

_{k=1}

_{k}

_{n}

*It remains to prove that regular p-variation of b* _{n} *and ν(A* _{+} ) + *ν(A* _{−} ) = 0 imply

_{n}

_{−}

### X *n* *k=1*

*hy, X* _{k} */b* _{n} *i −→*

_{k}

_{n}

*P* *0.* (20)

### This can be done in various ways. One can use, for example *the normal convergence criterion (with the limit δ* _{0} *= N (0, 0))* developed in Jakubowski and Szewczak (1991) together with the estimates of truncated moments given in Denker and Jakubowski *(1989). Less formal is the following procedure. Take {Y* _{k} ^{c} *} to be* independent, identically distributed and such that

_{k}

^{c}

### X *n* *k=1*

*Y* _{k} ^{c} */b* _{n} *−→*

_{k}

^{c}

_{n}

*D* *Stab (p, κ* _{(c,c)} *),*

_{(c,c)}

*where κ* _{(c,c)} is the same as in (7). By the corresponding one- *dimensional theorem, we have for x* _{n} increasing slowly enough

_{(c,c)}

_{n}

*x* ^{p} _{n} *P*

^{p}

_{n}

###

### X *n* *k=1*

*Y* _{k} ^{c} */b* *n* *> x* *n*

_{k}

^{c}

###

### *→ c,* as well as

*x* ^{p} _{n} *P*

^{p}

_{n}

###

### X *n* *k=1*

*Y* _{k} ^{c} */b* _{n} *< −x* _{n}

_{k}

^{c}

_{n}

_{n}

###

### *→ c.*

### It is now a matter of simple manipulations to deduce that we have also

*x* ^{p} _{n} *P*

^{p}

_{n}

###

### X *n* *k=1*

*(Y* _{k} ^{c} *+ hy, X* _{k} *i)/b* _{n} *> x* _{n}

_{k}

^{c}

_{k}

_{n}

_{n}

###

### *→ c,*

### (and similarly for the left-hand tails). Since Condition B _{1} is ob- *viously satisfied for the sequence {Y* _{k} ^{c} *+ hy, X* _{k} *i}, we obtain*

_{k}

^{c}

_{k}

### X *n* *k=1*

*(Y* _{k} ^{c} *+ hy, X* _{k} *i)/b* _{n} *−→*

_{k}

^{c}

_{k}

_{n}

*D* *Stab (p, κ* _{(c,c)} *).*

_{(c,c)}

*Letting c & 0 we obtain (20).*

*2. PROBABILITIES OF LARGE DEVIATIONS IN IR* ^{d} It follows from the proof of Theorems 3 and 4 that instead of Condition LD _{d} as it stands in (12) one can restrict the attention to verifying whether

^{d}

_{d}

*x* ^{p} _{n} *P (S* *n* */b* *n* *∈ x* *n* *A) −→ ν(A),* (21) *for much smaller class of sets A ∈ B* ^{d} than the whole ring of *bounded away from zero sets of ν-continuity. For example it is* *enough to consider sets A* ^{y} _{±} *, y ∈ IR* ^{d} , defined by (18) or “radial”

^{p}

_{n}

^{d}

^{y}

_{±}

^{d}

### sets described in (11). However, the problem does not seem to be easier after simplification of such kind.

### Fortunately, there are methods of essential reduction of (21) to *problems depending on properties of joint distributions of a fixed* *finite number of random variables X* 1 *, X* 2 *, . . . , X* *m* . These meth- ods were discussed in great detail in Jakubowski (1997) for the *case d = 1. Here we shall describe only basic steps in derivation* of their multidimensional versions.

### In all considerations the following generalization of the well-

### known Bonferroni’s inequality is crucial.

*Lemma 5 (Lemma 3.2, Jakubowski, 1997). Let Z* _{1} *, Z* _{2} *, . . . be* *stationary random vectors taking values in a linear space (E, B* _{E} *).*

_{E}

*Set T* _{0} *= 0, T* _{m} = ^{Pm} _{j=1} *Z* _{j} *, m ∈ IN. If U ∈ B* _{E} *is such that 0 /* *∈ U,* *then for every n ∈ IN and every k ∈ IN, k ≤ n, the following in-* *equality holds:*

_{m}

^{Pm}

_{j=1}

_{j}

_{E}

*|P (T* _{n} *∈ U) − n(P (T* _{k+1} *∈ U) − P (T* _{k} *∈ U))|*

_{n}

_{k+1}

_{k}

*≤ 3kP (Z* _{1} *6= 0) + 2* ^{X}

*1≤i<j≤n*
*j−i>k*

*P (Z* _{i} *6= 0, Z* _{j} *6= 0).* (22)

_{i}

_{j}

### To be applied effectively, inequality (22) requires that random *vectors Z* *n* with great probability take value 0 and that clusters *of nonzero values are essentially of short length (i.e. of size k).*

*This can be achieved by subtracting from X* _{k} */b* _{n} their truncation *around origin (in such a way that the total sum S* _{n} */b* _{n} is little per- turbed - typical property of heavy-tailed random elements), and imposing mixing conditions which guarantee a kind of asymp- totic independence of remaining “big” parts of components. The whole procedure is laborious and completely analogous to the one-dimensional case, hence we refer to Jakubowski (1997) for details.

_{k}

_{n}

_{n}

_{n}

*We shall consider three cases of particular interest: ψ-mixing,* *m-dependent and φ-mixing sequences (for definitions see Bradley* and Bryc, 1985, or Jakubowski, 1993) satisfying the following

### “usual conditions”:

*U0. X* _{1} *, X* _{2} *, . . . are strictly stationary random vectors.*

*U1. {b* _{n} *} is a 1/p-regularly varying sequence for some p, 0 < p <*

_{n}

### 2.

*U2. For some K* _{0} *< +∞*

*n∈IN* sup sup

*x>0* *x* ^{p} *· n · P (kX* 1 *k > x · b* *n* *) ≤ K* 0 *.* (23)

^{p}

*U3. If p = 1, then the law of X* _{1} *, L(X* _{1} ), is symmetric.

*U4. If 1 < p < 2, then EX* 1 = 0.

*Theorem 6 Suppose {X* _{k} *} is exponentially ψ-mixing (i.e. ψ(n)*

_{k}

*≤ Kη* ^{n} *, n = 1, 2, . . . , for some K > 0 and 0 < η < 1), and such* *that*

^{n}

*ψ(1) < +∞.* (24)

*Then for all x* _{n} *increasing slowly enough, as n → +∞,*

_{n}

*x* ^{p} _{n} *|P (S* _{n} */b* _{n} *∈ x* _{n} *A) − nP (X* _{1} */b* _{n} *∈ x* _{n} *A)| → 0,* (25) *for all A ∈ B* ^{d} *, A 63 0. In particular, nP (X* _{1} *∈ b* _{n} *· A) → ν(A)* *implies*

^{p}

_{n}

_{n}

_{n}

_{n}

_{n}

_{n}

^{d}

_{n}

*x* ^{p} _{n} *P (S* _{n} */b* _{n} *∈ x* _{n} *A) → ν(A).* (26) *Theorem 7 Let {X* _{k} *} be m-dependent. Then for all x* _{n} *increas-* *ing slowly enough*

^{p}

_{n}

_{n}

_{n}

_{n}

_{k}

_{n}

*x* ^{p} _{n} *|P (S* _{n} */b* _{n} *∈ x* _{n} *A)* (27)

^{p}

_{n}

_{n}

_{n}

_{n}

*−n (P (S* _{m+1} */b* _{n} *∈ x* _{n} *· A) − P (S* _{m} */b* _{n} *∈ x* _{n} *· A))| → 0,* *for all A ∈ B* ^{d} *, A 63 0. In particular, if*

_{m+1}

_{n}

_{n}

_{m}

_{n}

_{n}

^{d}

*n (P (S* _{m+1} *∈ b* _{n} *· A) − P (S* _{m} *∈ b* _{n} *· A)) → ν(A),* (28) *as n → +∞, then*

_{m+1}

_{n}

_{m}

_{n}

*x* ^{p} _{n} *P (S* _{n} */b* _{n} *∈ x* _{n} *A) → ν(A).* (29) *Theorem 8 Suppose {X* _{k} *} is exponentially φ-mixing Then for* *all x* _{n} *increasing slowly enough*

^{p}

_{n}

_{n}

_{n}

_{n}

_{k}

_{n}

### lim sup

*m* lim sup

*n* *x* ^{p} _{n} *|P (S* _{n} */b* _{n} *∈ x* _{n} *A)* (30)

^{p}

_{n}

_{n}

_{n}

_{n}

*− n (P (S* _{m+1} */b* _{n} *∈ x* _{n} *· A) − P (S* _{m} */b* _{n} *∈ x* _{n} *· A))| = 0,* *for all A ∈ B* ^{d} *, A 63 0. In particular, if for each m ∈ IN we have*

_{m+1}

_{n}

_{n}

_{m}

_{n}

_{n}

^{d}

*nP (S* _{m} *∈ b* _{n} *· A) → ν* _{m} *(A)* (31)

_{m}

_{n}

_{m}

*and, as m → ∞,*

*ν* _{m+1} *(A) − ν* _{m} *(A) → ν(A),* (32) *then*

_{m+1}

_{m}

*x* ^{p} _{n} *P (S* _{n} */b* _{n} *∈ x* _{n} *A) → ν(A).* (33) Remark 9 Theorems 6-8 can be used as tools for proving limit theorems based on Theorems 3 and 4. For example Theorem 6 leads to a result similar to that of Davis (1983) (obtained by purely one-dimensional methods). Theorem 7 allows prov- *ing results for m-dependent stationary random vectors due to* Jakubowski and Kobus (1989) and Kobus (1995) (originally ob- tained by the point processes technique). Theorem 8 corresponds to Theorem 3.9 in Jakubowski (1997), and gives a counterpart to the early Ibragimov’s central limit theorem (Ibragimov, 1962).

^{p}

_{n}

_{n}

_{n}

_{n}

### Remark 10 Let us notice that in formulas (27), (29) and (33) *probabilities of large deviations “in direction” of the set A does* *not depend on values of random variables outside of the “direc-* *tion” A. This fact is far from being obvious! In particular, in* the list of “usual conditions” we did not assume regularity in all

### “directions” (U2 says that there is no “dominating direction”), *and so the whole sum S* _{n} */b* _{n} may be divergent while Condition LD _{d} *holds for some family of sets A.*

_{n}

_{n}

_{d}

### Remark 11 In Davis and Hsing (1995), under less general con-

### ditions, an interesting probabilistic representation is given for

*constants c* _{+} *and c* _{−} appearing in Theorems 1 and 2. This rep-

_{−}

### resentation is expressed in terms of functionals of certain point

*processes naturally associated with the sequence {X* *k* *}. Since the*

### structure of the multidimensional limit law is more complicated

*than in the case d = 1, it would be interesting to extend Davis*

### and Hsing’s results and explain the mechanism of generating the

### limiting L´evy measure in Condition LD _{d} .

_{d}