An asymmetric Marcinkiewicz–Zygmund LLN for random fields

Statistics and Probability Letters 79 (2009) 1016–1020

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An asymmetric Marcinkiewicz–Zygmund LLN for random fields Allan Gut a,∗ , Ulrich Stadtmüller b a

Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden

b

Ulm University, Department of Number Theory and Probability Theory, D-89069 Ulm, Germany

article

info

Article history: Received 2 October 2008 Received in revised form 8 December 2008 Accepted 9 December 2008 Available online 24 December 2008 MSC: primary 60F15 60G50 60G60 secondary 60F05

a b s t r a c t The classical Marcinkiewicz–Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469–482] to random fields. Therein all indices have the same power in the normalization. Looking into some weighted means of random fields, such as Cesàro summation, it is of interest to generalize these laws to the case where different indices have different powers in the normalization. In this paper we give precise moment conditions for such laws. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Let X , {Xk , k ≥ 1} be i.i.d. random variables with partial sums {Sn , n ≥ 1}. The classical Marcinkiewicz–Zygmund strong law of large numbers (Marcinkiewicz and Zygmund, 1937) (see also Gut (2007), Theorem 6.7.1) reads as follows. Theorem 1.1. Let 0 < r < 2, and suppose that X , X1 , X2 , . . . are independent, identically distributed random variables. If E |X |r < ∞, and E X = 0 when 1 ≤ r < 2, then Sn

a.s.

n1/r

→ 0 as n → ∞.

Conversely, if almost sure convergence holds as stated, then E |X |r < ∞, and E X = 0 when 1 ≤ r < 2. Now, let, for d ≥ 2, Zd , be the positive integer d-dimensional lattice with coordinatewise partial ordering ≤. The following multiindex version of the Marcinkiewicz–Zygmund strong law was given in Gut (1978). Theorem 1.2. Let 0 < r < 2, and suppose that X , {Xk , k ∈ Zd } are i.i.d. random variables with partial sums Sn = n ∈ Zd . If E |X |r (log |X |)d−1 < ∞, and E X = 0 when 1 ≤ r < 2, then +

Sn

|n|

1/r

a.s.

→ 0 as n → ∞.

Conversely, if almost sure convergence holds as stated, then E |X |r (log+ |X |)d−1 < ∞, and E X = 0 when 1 ≤ r < 2.

∗

Corresponding author. Tel.: +46 18 471 3182; fax: +46 18 471 3201. E-mail addresses: [email protected] (A. Gut), [email protected] (U. Stadtmüller). URLs: http://www.math.uu.se/∼allan (A. Gut), http://www.mathematik.uni-ulm.de/matheIII/members/stadtmueller/stadtmueller.html (U. Stadtmüller). 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.12.006

P

k≤ n

Xk ,

A. Gut, U. Stadtmüller / Statistics and Probability Letters 79 (2009) 1016–1020

1017

Qd

Here |n| = k=1 nk and n → ∞ means min1≤k≤d nk → ∞, that is, all coordinates tend to infinity. Also, throughout the paper, log+ x = max{1, log x}. We wish to generalize the multiindex version so that instead of normalizing with |n|1/r we normalize with different powers for different coordinates. Looking into Cesàro means for random fields (see Gut and Stadtmüller (2008)) it is of interest to control these sums. The main results are presented in the following section followed by a section with proofs. For convenience we will use the parametrization α = 1/r, i.e., 1 ≤ r < 2 is translated into 1/2 < α ≤ 1. 2. Main results In order to formulate our main results let |nα | =

P Sn

|nα |

=

Qd

k=1

α

nk k . More precisely, we wish to examine the asymptotics for

Xk

k≤n

|nα |

,

where w.l.o.g. 1/2 < α1 = · · · = αp < αp+1 ≤ · · · ≤ αd ≤ 1,

(2.1)

and where p is some integer between 1 and d. In case p = d the situation reduces to that of Theorem 1.2. Our main result is as follows. Theorem 2.1. Assume the parameter constellation (2.1). Then E |X |1/α1 (log+ |X |)p−1 < ∞, and E X = 0 imply that Sn

|nα |

a.s.

→ 0 as n → ∞.

(2.2)

Conversely, (2.2) implies that E |X |1/α1 (log+ |X |)p−1 < ∞ and that E X = 0. Secondly we give a result on convergence in probability, which we need for e.g. desymmetrization, where a weaker moment assumption should suffice. Here only the number of indices but not the order of the index set is important. The following result is a slight extension of a result of Le Van Thanh in Le Van (2005). Theorem 2.2. Suppose that 1/2 ≤ α1 ≤ α2 ≤ · · · ≤ αd ≤ 1 with αd > 1/2. If E |X |1/α1 < ∞ and E X = 0, then

|Sn | |Sn | ≤ → 0 in L1/α1 and in probability as n → ∞. |nα | |n|α1 Actually it suffices here that max{j:αj >1/2} {nj } → ∞.

(2.3)

For the case 1/2 < α1 < 1 we have the following somewhat stronger result. Theorem 2.3. Suppose that 1/2 < α1 ≤ α2 ≤ · · · ≤ αd < 1. If n P (|X | > nα1 ) → 0 as n → ∞

(2.4)

then E |X | < ∞, and if E X = 0, then Sn

|nα |

p

→ 0 as max{nj } → ∞. j

Remark 2.1. Note that the Feller-type condition (2.4) is somewhat weaker than demanding that E (|X |1/α1 ) < ∞. Remark 2.2. The case maxj {nj } → ∞ is not relevant in Theorem 2.1 since the result there depends on the structure of the index set. The results so far show that the case when one or several α ’s are equal to 1/2 is special. Indeed, in the most extreme case when α1 = · · · = αd = 1/2 there is obviously no convergence in probability in view of the CLT. Let us therefore discuss the boundary cases w.r.t. a.s. convergence in dimension d = 2 in more detail. Again, if α1 = α2 = 1/2 then we are in the domain of the CLT and the pointwise sequences are a.s. unbounded. What happens if only α1 = 1/2? Here the following situation occurs. Theorem 2.4. If 1/2 = α1 < α2 ≤ 1 and, E X 2 < ∞ and E X = 0, then the following hold: Sm,n

p

m1/2 nα2 lim sup m,n→∞

√

→ 0 as m, n → ∞, Sm,n m1/2 nα2

Sm,n

m log log m nα2

(2.5)

= ∞ a.s.,

(2.6)

a.s.

→ 0 as m, n → ∞,

(2.7) log+ |X |

where for the last result we assume the slightly stronger moment condition E (X 2 log+ log+ |X | ) < ∞.

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A. Gut, U. Stadtmüller / Statistics and Probability Letters 79 (2009) 1016–1020

Remark 2.3. If in (2.7) we replace log log m by a function f (m) = o(log log m) then the random field is again a.s. unboundedly oscillating. For the proof, arguments similar to those for (2.6) can be used. 3. Proofs Proof of Theorem 2.1. For the following we define the random variables Yn = Xn 1{|Xn | ≤ |nα |}. Then,

X

X

P (Xn 6= Yn ) =

n

∞ X

P (|Xn | > |nα |) =

ν=1

n

∞ X

=

P (|Xn | > ν α1 )

X

1

αp+1 /α1 α /α n1 ···np np+1 ···nd d 1 =ν

P (|Xn | > ν α1 )∆g (ν) ≤

∞ X

g (ν)P (ν α1 < |X | ≤ (ν + 1)α1 ),

ν=0

ν=1

where

X

g (ν) :=

ν(log ν)p−1 (ν − 1)!

1∼c

αp+1 /α1 α /α ···nd d 1 ≤ν n1 ···np ·np+1

as ν → ∞

with a suitable constant c > 0 (see Lemma 3 in Stadtmüller and Thalmaier (2008) or Lemma 15 in Thalmaier (2008)) and ∆g (ν) = g (ν) − g (ν − 1). Now standard arguments show that the sum is finite iff E (|X |1/α1 (log+ |X |)p−1 ) < ∞. Hence we restrict our attention to the random variables Yn . Now, with β` = α` /α1 > 1 for p + 1 ≤ ` ≤ d , the function f (ν) :=

X

1∼

n1 ···np ≤ν

ν(log ν)p−1 (ν − 1)!

as ν → ∞

(see again Lemma 3 in Stadtmüller and Thalmaier (2008)), with differences ∆f (ν) = f (ν) − f (ν − 1) = find that αp+1

X Var (Yn ) n

|nα |2

∞ X

=

∆f (ν)

ν,np+1 ,...,nd =1

≤

ν

2α1

∞ X

X

2αp+1 np+1

·

···

2α nd d

2αp+1 np+1

j =1

···

2α nd d

∆f (ν)

X

2α

2αd

ν 2α1 · np+p1+1 · · · nd

∆f (ν)

X

j X

j =1

i =1

αp+1

∞ X

α

ν α1 np+1 ···nd d

1

ν,np+1 ,...,nd =1

j2 P (j − 1 < |X | ≤ j)

j =1 αp+1

∞ X

≤c

E (X 2 1{j − 1 < |X | ≤ j})

αp+1 α ν α1 np+1 ···nd d

1

2α ν,np+1 ,...,nd =1 ν 1 ·

α

ν α1 np+1 ···nd d

1

! i P (j − 1 < |X | ≤ j)

α

ν α1 np+1 ···nd d

1

X

iP (|X | > i) ∆f (ν) 2α 2α ν 2α1 · np+p1+1 · · · nd d i =1  αp+1 α np+1 ···nd d ∞ ∞ X X 1  X 1/α1 −1 2−1/α1 ≤c i i P (| X | > i ) ν −2α1 ∆f (ν)  2αp+1 2 αd n · · · n np+1 ,...,nd =1 p+1 i =1 ν=1 d  ∞ ∞ X X  + iP (|X | > i) ν −2α1 ∆f (ν)

≤c

ν,np+1 ,...,nd =1

αp+1 α i=np+1 ···nd d +1

αp+1

α

ν=(i/np+1 ···nd d )1/α1

 ≤c

∞ X

1

2αp+1 np+1 ,...,nd =1 np+1

···

2α nd d

 αp+1 α 2−1/α1 (np+1 · · · nd d )

αp+1 α np+1 ···nd d

X

i1/α1 −1 P (|X | > i)

i =1

 +

∞ X αp+1 α i=np+1 ···nd d +1

≤c

iP (|X | > i)

∞ X np+1 ,...,nd =1

αp+1 α ν=(i/np+1 ···nd d )1/α1

∞ X

1 βp+1

∞ X

β

np+1 · · · nd d i=1

 (ν −2α1 − (ν + 1)−2α1 ) f (ν)

i1/α1 −1 (log i)p−1 P (|X | > i),

P

n1 ···np =ν

1, we

A. Gut, U. Stadtmüller / Statistics and Probability Letters 79 (2009) 1016–1020

1019

which is finite iff E (|X |1/α1 (log+ |X |)p−1 ) < ∞, once again by Lemma 3 in Stadtmüller and Thalmaier (2008), since βk > 1 for p + 1 ≤ k ≤ d. In order to apply the multiindex Kolmogorov’s convergence criterion (see e.g Gabriel (1977)) we next show that

X E (Y ) X E (X 1{|X | > |nα |}) n = n |nα | n |nα | ∞ ∞ X X 1 ∆ g (ν) jα1 P ((j − 1)α1 < |X | ≤ jα1 ) α1 ν j=ν ν=1

≤

∞ X

≤

jα1 P ((j − 1)α1 < |X | ≤ jα1 )

j =1

j X 1 ∆g (ν), α1 ν ν=1

which is again finite by our moment assumption. Hence Kolmogorov’s convergence criterion applies and

Yn n |nα | ,

P

and thus

Xn n |nα |

P

converge almost surely. Finally, by the multiindex Kronecker lemma (cf. Moore (1966) for the necessary multiindex partial summation formula) we conclude that Sn /|nα | → 0 a.s. For the converse, we note that Sn

a.s.

|nα |

→ 0 as n → ∞

Xn /|nα | → 0

H⇒

as n → ∞

H⇒

X

P (|X | > |nα |) < ∞,

n

yielding the desired moment condition E (|X | that E X = 0. 

1/α1

)(log |X |) +

p−1

< ∞ (cf. Stadtmüller and Thalmaier (2008)) and, obviously, |S |

Proof of Theorem 2.2. Since the inequality in the statement is trivial we only have to prove convergence of |n|nα1 . In case α1 = 1/2 we may simply compute the variance which is bounded by 2αp+1 −1

E (X 2 )/(np+1

2αd −1

· · · nd

),

and tends to zero as max{np+1 , . . . , nd } → ∞. For α1 > 1/2 the Pyke–Root inequality (see Pyke and Root (1968)) tells us that

 E

|Sn | |n|α1

1/α1 ≤

o(|n|)

|n|

→ 0 as |n| → ∞,

which, in turn, establishes L1/α1 -convergence, and, hence, in particular, also convergence in probability. Proof of Theorem 2.3. Define have P

Ykn

α

= Xk 1{|Xk | ≤ |n |} and µn = |n|E (

Ykn

). Then, by the truncated Chebyshev inequality, we

  n X Sn − µn 1 >ε ≤ Var (Ykn ) + |n|P (|X | > |nα |) = In + IIn . |nα | |nα |2 ε 2 k=1

Note that IIn ≤ |n|P (|X | > |n|α1 ) → 0 as n → ∞ by assumption. Next,

|n| In ≤ |nα |2 ≤

≤

≤

≤

α /α α /α n1 n2 2 1 ···nd d 1

X

|

n2α−1

α /α α /α n1 n2 2 1 ···nd d 1

X

|

|n2α−1 |

|

X

j X

j =1

i=1

α /α α /α n1 n2 2 1 ···nd d 1

c

j2α1 P ((j − 1)α1 < |X | ≤ jα1 )

j =1 α /α α /α n1 n2 2 1 ···nd d 1

c

n2α−1

E (X 2 1{(j − 1)α1 < |X | ≤ jα1 })

j=1

1

X

|

! i

2α1 −1

P ((j − 1)α1 < |X | ≤ jα1 )

i2α1 −2 · i P (|X | > iα1 )

i=1

1−α /α c n2 2 1 2α −1 2α −α /α n1 1 n2 2 2 1

1−αd /α1

· · · nd

2α −α /α nd d d 1

··· → 0 as max{nj } → ∞, j

α /α α /α n1 n2 2 1 ···nd d 1

X i =1



i2α1 −2 · i P (|X | > iα1 )

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A. Gut, U. Stadtmüller / Statistics and Probability Letters 79 (2009) 1016–1020

since we apply – up to a bounded or asymptotically vanishing factor – a regular mean to a null sequence (cf. Gut (2007), Lemma A.6.1). If in addition α1 < 1 then, condition (2.4) implies that E (|X |) < ∞, since ∞ X

P (|X | > n) =

n =1

∞ X

1

· n1/α1 P (|X | > n) < ∞,

n1/α1 n =1

and, moreover, that

|n| |E (X 1{|X | ≤ |n|})| /|nα | = |n| |−E (X 1{|X | > |n|})| /|nα | ≤ |n|E (|X |1{|X | > |n|})/|nα | → 0 as |n| → ∞.  Proof of Theorem 2.4. The first result was just shown. For the next one we consider the subsequences mk = k and d

nk = log3 k := log log log k. Then, with the i.i.d. random variables Zk = X and partial sums Tn = equivalently, consider

√

T`k

`k log2 `k

·

Pn

k=1

Zk , we may,

(log2 `k )1/2 (log3 `k )α2 −1/2

with `k = k log3 k . This sequence oscillates unboundedly as k → ∞ by the law of iterated logarithm for any α2 . Note that in the LIL the behaviour along the subsequence `k is the same as that of the full sequence. The third result follows from the LIL for random arrays by Wichura (1973), which under the given moment assumption yields lim sup √ m,n→∞

Smn mn log log mn

a.s.

= Var (X ).

Now, since log log m log log mn

n2α2 −1 → ∞ as m, n → ∞,

the desired conclusion follows.



Acknowledgements The work in this paper has been supported by Kungliga Vetenskapssamhället i Uppsala. Their support is gratefully acknowledged. In addition, the second author would like to thank his partner Allan Gut for the great hospitality during two wonderful and stimulating weeks at the University of Uppsala. References Gabriel, J.-P., 1977. An inequality for sums of independent random variables indexed by finite dimensional filtering sets and its applications to the convergence of series. Ann. Probab. 5, 779–786. Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469–482. Gut, A., 2007. Probability: A Graduate Course. Springer-Verlag, New York, Corr. 2nd printing.. Gut, A., Stadtmüller, U., 2008. Cesàro summation for random fields. Report U.U.D.M. 2008:39, Uppsala University. Le Van, Thanh, 2005. On the Lp -convergence for multidimensional arrays of random variables. Int. J. Math. Sci. 8, 1317–1320. Marcinkiewicz, J., Zygmund, A., 1937. Sur les fonctions indépendantes. Fund. Math. 29, 60–90. Moore, C.N., 1966. Summable Series and Convergence Factors. Dover, New York. Pyke, R., Root, D., 1968. On convergence in r-mean for normalized partial sums. Ann. Math. Statist. 39, 379–381. Stadtmüller, U., Thalmaier, M., 2008. Strong laws for delayed sums of random fields. Preprint, University of Ulm. Thalmaier, M., 2008. Grenzwertsätze für gewichtete Summen von Zufallsvariablen und Zufallsfeldern. Dissertation, University of Ulm. Wichura, M.J., 1973. Some Strassen-type law of the iterated logarithm for multiparameter stochastic processes with independent increments. Ann. Probab. 1, 272–296.