Compound Poisson (CP) approximations have many applications and are one of the popular fields of research in modern probability theory, see, for example, [2, 3, 8, 9, 15, 23, 24] and the references therein. Research papers in this field are numerous. However, the absolute majority of results are proved for one-dimensional random variables. The situation is very different for random vectors (r.v.s), where results are a few and usually obtained only for Kolmogorov, Prokhorov, or local metrics, see [10, 12, 27] and the references therein. Note that, for sequences of r.v.s, a significant part of multidimensional research is devoted to normal approximation [4, 6] or behavior of tails of distributions [16]. We are primarily interested in triangular array of vectors.

In this paper, for sums of lattice identically distributed symmetric r.v.s, we adapt Le Cam’s convolution approach and properties of metrics to prove results for total variation. Apart from the general case, when no moment assumptions are imposed, we consider mixtures of distributions concentrated on separate lines with finite analogues of variances, demonstrating how known one-dimensional results can be applied to obtain the multidimensional estimates.

We introduce necessary notation. The sets of all integers and natural numbers are denoted respectively by Z and N. By M d and F d ⊂ M d we denote the set of all finite (signed) d-dimensional measures and the set of all d-dimensional distributions, respectively. Further on d < ∞ is some fixed natural number. Let I a → ∈ F d be the distribution concentrated at a → ∈ R d , I = I 0 → , where 0 → = ( 0 , 0 , … , 0 ) ∈ R d . Let M , V ∈ M d , then their convolution, for any d-dimensional Borel set A is defined as ( M ∗ V ) A = ∫ R d M A − x → V d x → .

By L ( X → ) we denote the distribution of a r.v. X →. Note that if S n = X → 1 + X → 2 + ⋯ + X → n , where all r.v.s. are independent identically distributed (i.i.d.) and L ( X → 1 ) = F, then L ( S n ) = F ∗ n , where power is understood in the convolution sense. In this paper we prefer measure notation over r.v.s, since the main tools used for the proofs are properties of norms and convolutions.

Exponential measure for M ∈ M d is defined as exp ( M ) = ∑ j ⩾ 0 M ∗ j / j !, where M ∗ 0 = I.

Compound Poisson distribution with parameter λ > 0 and compounding distribution F ∈ F d is defined as exp ( λ ( F − I ) ). Note that CP distribution can be expressed also as a distribution of random sum Y → = ∑ j = 0 π λ X → j , where X → , X → 1 , X → 2 , … are i.i.d. r.v.s independent from Poisson random variable π λ and L ( X → ) = F. CP distribution exp ( F − I ) is called accompanying distribution to F ∈ F d . If F ∈ F d , but λ < 0, then exp ( λ ( F − I ) ) is called signed compound Poisson measure (SCP). All CP distributions are infinitely divisible.

For F , G ∈ F d total variation distance and total variation norm are defined as d T V ( F , G ) : = sup B F B − G B = : 1 2 ‖ F − G ‖ T V , where supremum is taken over all d-dimensional Borel sets. Total variation norm can also be defined via Hahn decomposition, see [8], p. 228. For F , G ∈ F d with supp F, supp G ⊂ Z d ‖ F − G ‖ T V = ∑ m → ∈ Z d F m → − G m → .

To make proofs more concise the same symbol C is used to denotes all positive constants. If ambiguities can arise we supply C with indices. Symbol C ( N ) denotes quantity, which depends only on N. Notation f ( n ) = O ( n − k ) means that f ( n ) n k is bounded by some C for all n. Multiplication is superior to division. For any x ∈ R let ⌊ x ⌋ denote the integer part of x.

General CP approximations for independent symmetric vectors in Kolmogorov and Prokhorov metrics were investigated in [10, 28] (see also references, therein). In this paper, we consider stronger total variation metric. Multidimensional estimates in total variation for CP approximations are few. Dependent vectors were investigated in Theorem 6.8 [17]. Non-symmetric vectors were considered by Roos [20–22], who used the so-called Kerstan’s method. In [14], Kerstan’s method was applied for symmetric vectors concentrated on coordinate axes of Z d . Set e ¯ 1 = ( 1 , 0 , … , 0 ),…, e ¯ m = ( 0 , … , 0 , 1 , 0 , … , 0 ), e ¯ d = ( 0 , … , 0 , 1 ). Let X → 1 , … , X → n be independent r.v.s taking values in Z d and let for m = 1 , 2 , … , d , i = 1 , 2 , … , n: L ( X → i ) = q i I + ∑ m = 1 d p i , m F m , q i = I P ( X → i = 0 → ) , q i + ∑ m = 1 d p i , m = 1 , s u p p F m ⊂ k e ¯ m , k = ± 1 , ± 2 , … , F m k e ¯ m = F m − k e ¯ m , λ m = ∑ i = 1 n p i , m , σ m 2 = 2 ∑ k = 1 ∞ k 2 F m k e ¯ m , 0 ⩽ q i , p i , m ⩽ 1 , α = ∑ i = 1 n g ( 2 ( 1 − q i ) ) min 2 − 3 / 2 ∑ m = 1 d p i , m 2 λ m , ( ∑ m = 1 d p i , m ) 2 , g ( x ) = 2 ∑ s = 2 ∞ x s − 2 s ! ( s − 1 ) = 2 e x ( e − x − 1 + x ) x 2 , x ≠ 0 . Observe that in above setting coordinates of X → i are uncorrelated and, for example, I P ( X → i = ( 1 , 1 , 1 , … , 1 ) = 0. In [14] it was proved that if 2 α e < 1, σ m 2 < ∞, ( m = 1 , 2 , … , d), then (1) d T V ∏ i = 1 ∗ n L ( X → i ) , ∏ i = 1 ∗ n exp ( L ( X → i ) − I ) ⩽ 15.98 ( 1 − 2 α e ) 3 / 2 ∑ l = 1 d ( 1 + σ l ) ∑ m = 1 d 1 λ m 2 ∑ i = 1 n p i , m 2 .

If d , σ m , p i , m ≍ C, then the rate of accuracy in (1) is O ( n − 1 ).

In the case of i.i.d r.v.s. X → i ≡ X →, when p i , m ≡ p m and q = I P ( X → = 0 → ) ⩾ 4 / 5, it was proved that (2) d T V q I + ∑ m = 1 d p m F m ∗ n , exp n ∑ m = 1 d p m ( F m − I ) ⩽ 17.34 n ∑ m = 1 d 1 + σ m 2 , see [14], Theorem 3.2. Kerstan’s method has obvious restrictions. For (1) and (2) to hold, probabilities I P ( X → i = 0 → ) must be large. Consider, for example, the case d = 1, q i = p i = 1 / 2. Then 2 α e > 1 and (1) cannot be applied. Moreover, σ m can be large even if supp X → i is finite. The assumption that all coordinate vectors are uncorrelated is also very restrictive.

Note that any distribution with q = F { 0 → } > 0 can be written as F = q I + ( 1 − q ) V. From one-dimensional Poisson approximation to the binomial law it follows then that (3) d T V ( F ∗ n , exp ( n ( F − I ) ) ) ⩽ min ( 1 − q , n ( 1 − q ) 2 ) , see [8], p. 207 and (3.15) for details. However, (3) does not take into account the symmetry of F and requires q to be very close to 1, much closer than needed for (2).

First we formulate general facts about exponential measures. Let M , V ∈ M d , k ∈ N. We have exp ( M ) ∗ exp ( V ) = exp ( M + V ). From definition of exponential measure it follows that (23) e M = I + M + M ∗ 2 2 ! + ⋯ + M ∗ k k ! + M ∗ ( k + 1 ) k ! ∗ ∫ 0 1 e τ M ( 1 − τ ) k d τ . Let F ∈ F d , a > 0. The following well-known relations hold: ‖ F ‖ T V = 1, (24) ‖ M ∗ V ‖ T V ⩽ ‖ M ‖ T V ‖ V ‖ T V , ‖ exp ( M ) ‖ T V ⩽ exp ( ‖ M ‖ T V ) , (25) ‖ ( F − I ) ∗ k ∗ exp ( a ( F − I ) ) ‖ T V ⩽ ‖ ( F − I ) ∗ exp ( a ( F − I ) / k ) ‖ T V k , (26) ‖ ∑ j = 0 ∞ α j F ∗ j ‖ T V ⩽ ∑ j = 0 ∞ | α j | = ‖ ∑ j = 0 ∞ α j I j ‖ T V , where I j ∈ F 1 is one-dimensional distribution concentrated at j. Sometimes inequality (26) is formulated in terms of random sums, see Eq. (3.1), [25]. It allows to reformulate many one-dimensional results for multidimensional case. Next lemma exemplifies such reformulation.

Estimate (27) follows from Lemma 4.6 in [7] and (26). Estimate (28) follows from Proposition A.2.7 in [1], (26) and (25).

The following technical estimates follow from Eq. (31) in [13] and from [19] Lemma 6.

Next we present generalization of Lemma B.2 from [8].