Limit theorems for sums of random variables with mixture distribution

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Model 3G

pp. 1–8 (col. fig: nil)

Statistics and Probability Letters xx (xxxx) xxx–xxx

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Limit theorems for sums of random variables with mixture distribution Vladimir Panov International Laboratory of Stochastic Analysis and its Applications, National Research University Higher School of Economics, Shabolovka, 26, Moscow, 119049, Russia

article

a b s t r a c t

info

Article history: Received 24 January 2017 Received in revised form 16 June 2017 Accepted 21 June 2017 Available online xxxx

In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions of the sums due to various relations between these parameters. © 2017 Elsevier B.V. All rights reserved.

MSC: primary 60G50 secondary 60F05 60E07 Keywords: Mixture distribution Central limit theorem Stable distribution Phase transition

1

2 3 4 5 6 7 8

9

1. Introduction Theory of limit distributions for the sums of random variables is well-described in brilliant books by Ibragimov and Linnik (1971), Meerschaert and Scheffler (2001), Petrov (2012). Usually, the most interest is drawn to 2 classical models: a model of i.i.d. random variables and triangular arrays. For the first model, it is common to find non-degenerate laws, which can 1 appear as a limit of the sums (ξ1 + · · · + ξn )b− n + an with i.i.d. ξ1 , . . . , ξn , and some deterministic sequences an , bn . It is well-known that the set of limiting distributions in this case coincides with the class of stable distributions. In the second model, one considers an infinitesimal triangular array - a collection of real random variables {Znk , k = 1 . . . kn }, kn → ∞ as n → ∞, such that Zn1 , . . . , Znkn are independent for each n and satisfy the condition of infinite smallness sup P {|Znk | > δ} → 0,

k=1...kn 10 11 12 13 14 15 16

n→∞

(1)

for any δ > 0. For this model, it is known that only the infinitely divisible distributions can appear as the non-degenerate limit of sums Zn1 + · · · + Znkn − an with deterministic an , and moreover, for any infinitely-divisible distribution there exists a triangular array (Znk ) such that the sum Zn1 + · · · + Znkn converges to this distribution. Nevertheless, the analysis of the limiting distribution in models can be rather tricky. For instance, Ben Arous, et ∑particular n al. (2005) analysed the asymptotic behaviour of the sums i=1 et ξi , where ξ1 , . . . , ξn is an i.i.d. sequence of r.v. with regularly varying log-tail functions, and n and t simultaneously tend to infinity, provided that the speeds of growth of n and t are coordinated via the parameter λ = lim infn,t →∞ log(n)/log(E[et ξ ]). It turns out, that there exist two critical values of this E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2017.06.017 0167-7152/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

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parameter, λ1 < λ2 , below which the law of large numbers and the central limit theorem (respectively) break down. This result generalizes some previous findings related to the random energy model, which corresponds to the case when ξi are standard normal, see Bovier et al. (2002). In the current research, we derive similar results for completely different model, defined as a mixture of two distributions: the first distribution has light tails and the second is constructed by truncation of the distribution with heavy tails. More precisely, let F1 (·) be a distribution function corresponding to a probability distribution on R+ with the upper tail in exponential form, that is, F1 (x) = 1 − e

−λx

(1 + o(1)) ,

x → +∞,

F2 (x) = 1 − x

(1 + o(1)) ,

F2M (x) =

F2 (x)/F2 (M), 0,

3 4 5 6 7

9

x → +∞,

10

with α ∈ (0, 2). By F2M (x) denote the truncated distribution function F2 at level M:

{

2

8

for some λ > 0. Let F2 (·) be a distribution function corresponding to a heavy-tailed distribution with support on [1, ∞), −α

1

11

if x ≤ M ; if x > M .

12

Next, consider the mixture of these distributions, that is, the distribution with distribution function F (x) = (1 − ε )F1 (x) + ε F2M (x),

13

(2)

where ε ∈ (0, 1) is a mixing parameter, which is assumed to be small. The motivation of considering such mixture goes to the idea to model some ‘‘frequent events’’ by light-tail distributions, and ‘‘rare events’’ by truncated heavy-tailed. For instance, this idea is quite natural for modelling the claim amounts in insurance, see e.g, Rolski et al. (1999), or Embrechts et al. (1997). Another example comes from population dynamics by analysing the migration of species. In some models, it is assumed that for most species in the population, the distribution of migration is light-tailed, whereas for some small amount of species the distribution is heavy-tailed, see Whitmeyer and Yang (2016). In this research we consider the case when ε and M depend on n, and moreover, ε → 0 and M → ∞ as n grows. We focus on studying the fluctuations of sums of random variables drawn from the mixture model (2), and aim to characterize the limit laws depending on the relation between ε and M. Several problems of this type are considered in the paper by Grabchak and Molchanov (2014), where the parameter M as well as both distributions in (2), are fixed. As it is shown in Grabchak and Molchanov (2014), the complete asymptotic analysis can be done by taking into account that the distributions are in the domain of attractions of some stable random variables. In this case, it is clear that the limit law for the mixture can be determined by the relation between the normalizing sequences. Nevertheless, this methodology cannot be applied to our set-up, since the parameters ε and M simultaneously vary. The paper is organized as follows. In the next section we formulate our main results. It turns out (and is not surprising) that the cases α ∈ (0, 1) and α ∈ [1, 2) are essentially different, see Sections 2.1 and 2.2 respectively. The proofs are collected in Section 3. 2. Limit theorems

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

32

Assume that for any n ∈ N we have kn independent random variables Zn1 , . . . , Znkn with mixing distribution (2). In other words, Znk = (1 − Bnk ) Xnk + Bnk Ynk ,

k = 1 . . . kn ,

(3)

M F2 n

where Xn1 , . . . , Xnkn ∼ F1 , Yn1 , . . . , Ynkn ∼ , Bn1 , . . . , Bnkn are Bernoulli random variables with probability of success equal to εn , and all Xnk , Ynk , Bnk , k = 1 . . . kn are jointly independent for any n. γ1 −γ2 In what follows, with positive γ1 , γ2 , and aim to characterize the asymptotic behaviour of ∑kn we take Mn = n , εn = n the sum Sn := Z due to the relation between γ1 and γ2 . nk k=1 2.1. Case α ∈ (0, 1)

33 34

35 36 37 38 39

40

We start with the most interesting case, α ∈ (0, 1).

41

Theorem 2.1.

42

(i) Let γ1 , γ2 be such that

43

γ2 > (2 − α )γ1

or

n

αγ1 +γ2

kn

→ 0.

(4)

Then the central limit theorem holds, in the sense that Sn − kn E [Zn1 ]

√

kn Var(Zn1 )

d

−→ N (0, 1),

n → ∞.

44

45

46 47

Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

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1

(ii) Let γ1 , γ2 be such that nαγ1 +γ2

γ2 < (2 − α) γ1 and

2

3

3

kn

1−α/2

→ ∞ and

kn

→ 0.

nγ 2

(5)

Then Sn − kn E[Xn1 ]

√

4

kn Var(Xn1 )

d

−→ N (0, 1),

n → ∞.

5 6

(iii) Finally, let γ1 , γ2 be such that nαγ1 +γ2

7

8

kn

10

Sn − κn

βn

16

17 18 19 20 21 22

n → ∞,

⎧ kn E[Xn1 ], ⎪ ⎪ ⎨ cα κn = kn E[Xn1 ] + 1 − α βn , ⎪ ⎪ cα ⎩ βn , 1−α

13

14

d

−→ Fα,c ,

where Fα,c is an α -stable distribution on R+ , that is, an infinitely divisible distribution with the Lévy density s(x) = ( )1/α cx−1−α I {x > 0}, and without continuous part. More precisely, βn = c −1 kn n−γ2 , and

12

15

if k1n−α n−γ2 → 0, if k1n−α = nγ2 , if k1n−α n−γ2 → ∞.

Remark 1. Note that all conditions can be simplified in the partial case kn = n. In fact, condition (4), which guarantees the convergence as in the central limit theorem, is equivalent to

γ2 > (2 − α )γ1

{ } γ2 < min (2 − α )γ1 , 1 − αγ1 .

or

Condition (5), which leads to an analogue of the central limit theorem with another normalization, is equivalent to

γ1 > 1/2

γ2 ∈ (1 − α/2, (2 − α )γ1 ) .

and

Finally, condition (6) can be rewritten as

γ1 > 1/2

γ2 ∈ (max (1 − αγ1 , 0) , 1 − α/2) .

and

It would be a worth mentioning that the normalizing term in (ii) cannot be changed to essentially differs from the central limit theorem. In fact,

( 23

Var(Zn1 ) =

Var(Xn1 ) +

2α 2−α

n(2−α )γ1 −γ2

)

(1 + o(1)),

24

and hence Var(Zn1 ) ≍ Var(Xn1 ) if and only if γ2 > (2 − α )γ1 .

25

Theorem 2.2. Let γ1 , γ2 be such that

26

(6)

→ ∞.

nγ 2

Then for any c > 0, there exist sequences βn , κn such that

9

11

1−α/2

kn

and

→∞

γ2 > (1 − α )γ1

or

nαγ1 +γ2 kn

n→∞

→ 0.

√

kn Var(Zn1 ), and therefore (ii)

(7)

(8)

27 28

Then the law of large numbers holds, in the sense that Sn

29

kn E [Zn1 ]

p

−→ 1,

n → ∞.

30

31

32

33 34 35

Remark 2. Similarly to Remark 1, let us note that the condition (8) is equivalent to

γ2 > (1 − α )γ1

or

{ } γ2 < min (1 − α )γ1 , 1 − αγ1 .

Remark 3. In some cases there exists an analogue of Theorem 2.1 with another normalization, equal to kn E[Xn1 ]. More precisely, if γ1 , γ2 are such that

γ2 > 1 − α

and

γ2 ∈ (1 − α, (1 − α )γ1 ) ,

Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

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then

1

Sn

p

−→ 1,

nE [Xn1 ]

n → ∞,

2

provided kn = n.

3

Analogously to (7), the relation between E [Xn1 ] and E [Zn1 ] is given by

α

( E [Zn1 ] =

E [Xn1 ] +

n(1−α )γ1 −γ2

1−α

)

(1 + o(1)),

4

n → ∞,

(9)

and therefore E [Zn1 ] ≍ E [Xn1 ] if and only if γ2 > (1 − α )γ1 . Fig.∑ 1 illustrates the division of the area (γ1 , γ2 ) ∈ R+ × R+ into subareas with different asymptotic properties of the k sums kn=1 Xnk for the case kn = n. 2.2. Case α ∈ [1, 2)

7 8

10

Theorem 2.3. Let α ∈ [1, 2). Then

11

(i) for any positive γ1 , γ2 the law of large numbers holds, i.e., Sn

6

9

In this case, the limit laws are more simple. We formulate the results in the next theorem.

p

kn E [Zn1 ]

5

−→ 1,

12

n → ∞;

13 14

(ii) if γ1 , γ2 satisfy (5), then the central limit theorem holds, i.e., Sn − kn E [Zn1 ]

√

kn Var(Zn1 )

d

−→ N (0, 1),

15

n → ∞;

16 17

(iii) otherwise, if (5) is not fulfilled, then for any constant c > 0, Sn − kn E [Zn1 ]

βn

d

−→ Fα,c ,

18

n → ∞,

19

see Theorem 2.1(iii) for notations.

20

3. Proofs

21

We first prove the statements related to the law of large numbers (Theorem 2.2 and Theorem 2.3(i)), then the central limit theorems (Theorem 2.1(i) and Theorem 2.3(ii)), and afterwards we show results revealing other types of asymptotic behaviour of the sums (Theorem 2.1(ii), (iii) and Theorem 2.3(iii)). Proof of Theorem 2.2(i) and Theorem 2.3(i). 1. Denote

˜ Sn =

∑kn

Sn

E[Sn ]

−1=

k=1

(Znk − E [Znk ]) kn E [Zn1 ]

r

]

24

26

.

27

r

[

23

25

p

Our aim is to show that there exists a constant r > 1 such that E|˜ Sn | → 0 as n → ∞. This will imply that ˜ Sn −→ 0, and therefore the result will follow. Applying the Bahr–Esseen inequality for r ∈ (1, 2), see von Bahr and Esseen (1965), we get that

E |˜ Sn |

22

28 29 30

∑kn ≤ Cr

[ ] r k=1 E |Znk − E [Znk ]| (kn E [Zn1 ])r

= Cr k1n−r

Mn (r)

(E [Zn1 ])r

,

31

(10)

where Cr is some constant depending on r, and Mn (r) is the rth absolute central moment of Zn1 . The further analysis is consists in establishing the asymptotical behaviour of the numerator and denominator of the fraction in (10). 2. Note that for any n = 1, 2, . . .

E[Zn1 ] = (1 − εn ) · µ1 (1) + εn · µ2 (1), Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

32

33 34

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5

∑n

Fig. 1. Illustration of the asymptotic behaviour of the sums k=1 Xnk depending on γ1 and γ2 for the case α ∈ (0, 1) and kn = n. zone 1(red): both the central limit theorem and the law of large numbers hold (see Theorem 2.1(i) and Theorem 2.2); zone 2 (yellow): convergence to the standard normal distribution under normalization nE[Xn1 ] and the law of large numbers (see Theorem 2.1(ii) and Theorem 2.2); zone 3 (orange): convergence to the standard normal distribution under normalization nE[Xn1 ] and the analogue of the LLN with the same normalization (see Theorem 2.1(ii) and Remark 3); zone 4 (blue): convergence to stable distribution and the analogue of the LLN with normalization nE[Xn1 ] (see Theorem 2.1(iii) and Remark 3); zone 5 (purple): convergence to stable distribution (see Theorem 2.1(iii)); zone 6 (green): convergence to stable distribution and LLN (see Theorem 2.1(iii) and Theorem 2.2).

1

2

3

4

5

6

where µ1 (s) = E[(Xn1 )s ] < ∞ for any s > 0, and

µ2 (s) = E[(Yn1 )s ] =

α M max(s−α,0) (1 + o(1)) , |s − α| n ⎩α log(M ) (1 + o(1)) , n

⎧ ⎨

if s ̸ = α ; if s = α .

Therefore,

⎧( ) α 1−α ⎪ (1 + o(1)) , ⎪ ⎨ µ1 (1) + 1 − α εn Mn ( ) E[Zn1 ] = µ1 (1) + εn log(Mn ) (1 + o(1)) , ⎪ ⎪ ⎩ µ1 (1) (1 + o(1)) ,

if α < 1; (11)

if α = 1; if α > 1.

3. It holds for any r > 1

[⏐ ] [ ]⏐⏐r ⏐ Mn (r) = E ⏐⏐ (1 − Bn1 ) Xn1 + Bn1 Yn1 − E (1 − Bn1 ) Xn1 + Bn1 Yn1 ⏐⏐ {

7

r −1

≤4

} r

[

E (1 − Bn1 )

]

·E

r Xn1

[

]

+E

[

Brn1

]

·E

r Yn1

[

]

+ (1 − εn ) · (EXn1 ) + ε · (EYn1 ) r

r

{ 8

9

r −1

=4

r n

r

}

(1 − εn )µ1 (r) + εn µ2 (r) + (1 − εn ) (µ1 (1)) + ε (µ2 (1)) r

r

r n

r

.

Denote Dr = µ1 (r) + (µ1 (1))r and consider two cases: (a) if (α < 1, r > 1) or (1 < α ≤ r), then it holds r −1

Mn (r) ≤ 4 10

) α r −α εn Mn · (1 + o(1)) , Dr + r −α

(

where we use that εn µ2 (r) ≳ εnr (µ2 (1))r as n → ∞; (b) otherwise, if (1 < r < α ) or (α = 1, r > 1), then we have Mn (r) ≤ 4r −1 Dr (1 + o(1)) . Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

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4. To conclude the proof, we substitute the upper estimate for Mn (r) and (11) into (10). If α < 1, then

) + εn Mnr −α · (1 + o(1)) ≤˜ Cr · k1n−r ( )r (2) cr + εn Mn1−α · (1 + o(1)) (

[

r E |˜ Sn |

]

(1)

(1)

cr

(12)

(2)

with some constants cr , cr (depending on r) and a bounded function ˜ Cr . The asymptotic behaviour of the last fraction differs between the following two cases:

• εn Mn1−α → 0, that is, γ2 > (1 − α )γ1 . Then there exists r > 1 such that εn Mnr −α → 0 - in fact, one can take r = (γ2 − (1 − α )γ1 )γ1−1 + 1 > 1. Under this choice of r, the r.h.s. of (12) tends to 0, and therefore the law of large numbers holds for any (γ1 , γ2 ) s.t. γ2 > (1 − α )γ1 . • εn Mn1−α → ∞, that is, γ2 < (1 − α )γ1 . Then the right-hand side of (12) tends to 0 if and only if Mnα /(kn εn ) → 0, that is, iff nαγ1 +γ2 /kn → 0. [ ] r In other cases, α > 1 and α = 1, we can choose r ∈ (1, α ) and get that E |˜ Sn | ≲ k1n−r , and therefore the law of large numbers holds with any positive γ1 , γ2 . Proof of Theorem 2.1(i) and Theorem 2.3(ii). To prove these theorems, we check that the Lyapunov condition holds (see (27.16) from Billingsley (1995)): there exists δ > 0 such that

Ωn :=

Mn (2 + δ ) δ/2

kn (Var(Zn1 ))1+δ/2

→ 0,

as n → ∞.

1 2

3 4 5 6 7

8 9

10 11

12

The variance of Zn1 has the following asymptotical behaviour:

13

Var(Zn1 ) = (1 − εn ) µ1 (2) + εn µ2 (2) − (E[Zn1 ])2

14

( =

[ ] µ1 (2) − (µ1 (1))2 +

) α 2−α · (1 + o(1)) , εn Mn 2−α

n → ∞,

15

and the numerator of Ωn was already considered in the proof of Theorem 2.2(i). Therefore, c2 + ε

( Ωn ≤ c ·

δ/2 (

kn

)

· ( + o(1))

2+δ−α 1 n Mn 2−α 1+δ/2 n Mn

c3 + ε

· (1 + o(1))

)

,

16

n → ∞,

17

with some positive constants c , c2 , c3 . The rest of the proof follows the same lines as Step 4 in the proof of Theorem 2.2(i), see above. Proof of Theorem 2.1 (ii), (iii) and Theorem 2.3(iii). The proof is based on the following proposition, which is in fact a combination of Theorem 1.7.3 from Ibragimov and Linnik (1971), Theorem 3.2.2 from Meerschaert and Scheffler (2001), and a number of theorems given in Chapter IV from Petrov (2012). Proposition 3.1. Consider an infinitesimal triangular array {Znk , k = 1 . . . kn }, such that (1) is fulfilled. In what follows, we denote the distribution of Znk by µnk , and use the notation Sn := Zn1 + · · · + Znkn . The following statements hold. (1) If there exists a random variable Y and a sequence of real numbers an such that d

Sn − an −→ Y ,

n → ∞,

18 19

20 21 22

23 24

25

(13)

26

then Y has an infinitely divisible distribution; moreover, for any infinitely distribution Pinf there exists a triangular array

27

d

{Znk , k = 1 . . . kn } such that Sn −→ Pinf . (2) There exists a deterministic sequence an such that sequence Sn − an converges weakly to an infinitely divisible random variable Y with characteristic exponent 1

ψ (u) = iuµ − u2 σ 2 + 2

∫

eiux − 1 − iuxII {|x| ≤ 1} ν (dx),

( R/{0}

)

where (µ, σ , ν ) is a Lévy triplet, if and only if the following conditions are fulfilled: (a)

∑kn

k=1

µnk (A) → ν (A) for any A = (−∞, x) with x < 0 and any A = (x, +∞) with x > 0 such that ν (∂ A) = 0;

Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

28 29 30

31

32 33

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7

(b) moreover, for some τ > 0, lim lim sup

τ →0

{∫ kn ∑

n→∞

k=1

= lim lim inf τ →0 n→∞

an =

2

kn ∫ ∑ k=1

6 7

|x|<τ

x µnk (dx) |x|<τ

{∫ kn ∑

x µnk (dx) − 2

(∫

x µnk (dx)

)2 }

= σ2

(14)

|x|<τ

|x|<τ

k=1

)2 }

x µnk (dx) + o(1),

(15)

|x|<1

provided ν {x : |x| = 1} = 0.

3

5

(∫

If these conditions are satisfied, an may be chosen according to the formula

1

4

x µnk (dx) − 2

(3) There exists a deterministic sequence an such that sequence Sn − an converges weakly to a standard normal random variable Y if and only if the following conditions are fulfilled:

∑kn

{|Znk | > { x} → 0 as n → ∞(for any x > 0; ) } 2 ∫ ∑kn ∫ x2 µnk (dx) − µ (dx) = 1 for some τ > 0. If these conditions are satisfied, an may (b) limn→∞ k=1 |x|<τ |x|<τ nk (a)

k=1 P

be chosen according to (15).

8

9

10

11

Returning to our setup, we denote Fnk (x) = P {Znk ≤ βn x}, and first note that for any x ∈ (βn−1 , βn−1 Mn ) kn ∑

(1 − Fnk (x)) =

k=1

kn [ ∑

1 − (1 − εn ) F1 (βn x) − εn

k=1

F2 (βn x) F2 (Mn )

(

12

= kn (1 − εn ) (1 − F1 (βn x)) + kn εn 1 −

13

= kn (1 − εn )e

−λβn x

16 17

18

F2 (βn x)

)

F2 (Mn )

(1 + o(1)) + kn εn (βn x)−α (1 + o(1)) −kn εn Mn−α (1 + o(1)) .

14

15

]

Note that basically only 3 situations are possible. 1. nαγ1 +γ2 /kn → ∞and kn n−γ2 → ∞. In this case, under the choice βn = c −1 kn n−γ2

(

)1/α

with any constant c > 0, we

get kn ∑

(1 − Fnk (x)) → cx−α ,

∀ x ∈ R+ ,

k=1 1−α/2 −γ2

19

because kn εn βn−α → c, kn εn Mn−α → 0, and kn e−λβn x → 0 provided kn

20

fulfilled — in fact, for any δ > 0, it holds

21

1 sup P {|Znk | > δ} = k− n

k=1...kn

22

kn ∑

n

→ ∞. Moreover, the condition (1) is

(1 − Fnk (δ )) → 0.

k=1

Next, with any s ≥ 1,

∫

[

23

|x|<τ

∫

s

24

|x|<τ

]

s xs ˜ P1 (dx) = βn−s · E Xn1 (1 + o(1)) ,

x ˜ P2 (dx) =

α (1 + o(1)) , |s − α| −s αβn log(βn ) (1 + o(1)) ,

{ βn− min(α,s) τ max(s−α,0)

if α ̸ = s; if α = s

Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

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where ˜ P1 , ˜ P2 are the probability distributions of Xn1 /βn and Yn1 /βn resp. Therefore, if α < 1, the condition (14) reads as Gn :=

{∫ kn ∑

x µnk (dx) − 2

x µnk (dx)

|x|<τ

|x|<τ

k=1

)2 }

(∫

}

{ 2 + c εn βn−α τ 2−α − βn−1 E[Xn1 ] + Rn = kn βn−2 E Xn1

[

(

]

)2

(1 + o(1))

where

1

{ Rn =

C2 εn βn− min(1,α ) τ 1−α , C3 εn βn−1 log(βn ),

if α ̸ = 1, if α = 1,

2

(

1−α/2 −γ2

and c , C2 , C3 > 0. We conclude that if kn βn−2 = kn

n

)−2/α

→ 0, then limτ →0 limn→∞ [Gn ] = 0; otherwise the

last limit is infinite. At the same time, (15) yields for α ̸ = 1,

[ an = k n

1 − εn

βn

E [Xn1 ] +

εn βnmin(1,α)

] α + o(1). |1 − α|

(16)

For instance, if α < 1, then an =

kn E [Xn1 ]

βn

+

cα 1−α

3

+ o(1).

4

Otherwise, in the case α = 1,

[ an = kn

1 − εn

βn

E [Xn1 ] +

] εn kn E[Zn1 ] log(βn ) + o(1) = + o(1), βn βn

where we use (11). √ 1−α/2 −γ2 2. kn n → 0and nαγ1 +γ2 /kn → ∞. In this case, we take βn = kn Var(Xn1 ). Under this choice, the conditions (a) and (b) from Part 3 of Proposition 3.1 hold. The choice an = kn E[Xn1 ]/βn follows from (16). 3. nαγ1 +γ2 /kn → 0. It is easy to see that the infinite smallness condition (1) is not fulfilled. Note that this case was considered separately in Theorem 2.1(i). Acknowledgement The author is grateful to Prof. Stanislav Molchanov (UNC Charlotte, USA, and Higher School of Economics, Moscow, Russia) for the supervision of this research. References Ben Arous, G., Bogachev, L., Molchanov, S., 2005. Limit theorems for sums of random exponentials. Probab. Theory Related Fields 132 (4), 579–612. Billingsley, P., 1995. Probability and Measure, 3rd edition. Wiley and Sons. Bovier, A., Kurkova, I., Löwe, M., 2002. Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30 (2), 605–651. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer. Grabchak, M., Molchanov, S., 2014. Limit theorems and phase transitions for two models of summation of independent identically distributed random variables with a parameter. Theory Prob. Appl. 59 (2), 222–243. Ibragimov, I., Linnik, Yu., 1971. Independent and Stationary Sequences of Random Variables. Walters-Noordoff. Meerschaert, M., Scheffler, H.-P., 2001. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Vol. 321. John Wiley & Sons. Petrov, V., 2012. Sums of Independent Random Variables, Vol. 82. Springer Science & Business Media. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes for Insurance and Finance. In: Wiley Series in Probability and Statistics. J. Wiley. von Bahr, B., Esseen, C.-G., 1965. Inequalities for the r th absolute moment of a sum of random variables, 1 ≤ r ≤ 2. Annal. Math. Stat. 36 (1), 299–303. Whitmeyer, J., Yang, H., 2016. Baseline models of spatial population dynamics. J. Math. Sociol. 40 (2), 123–135.

Please cite this article in press as: Panov, V., Limit theorems for sums of random variables with mixture distribution. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.017.

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