Self-normalized laws of the iterated logarithm

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1 Journal of Statistical and Econometric Methods, vol.3, no.3, 2014, ISSN: print), online) Scienpress Ltd, 2014 Self-normalized laws of the iterated logarithm Igor Zhdanov 1 Abstract Stronger versions of laws of the iterated logarithm for self-normalized sums of i.i.d. random variables are proved. Keywords: Law of the iterated logarithm; stochastic compactness; self-normalization 1 Introduction The law of the iterated logarithm is one of the fundamental laws of the classical probability theory. The reader will find various versions of the law of the iterated logarithm reviewed in [1]. In the last decade, analogues of the law of the iterated logarithm were proved for sequences of self-normalized sums of i.i.d. random variables. The results are available in articles [2] and [3] as well as sources referenced in these articles. This paper presents stronger versions 1 Moscow State University named after M.V. Lomonosov, the Faculty of Computational Mathematics and Cybernetics. zhdanov.i@gmail.com Article Info: Received : June 21, Revised : August 5, Published online : August 15, 2014

2 146 Self-normalized laws of the iterated logarithm of some of the nown statements regarding the law of the iterated logarithm for sequences of self-normalized sums of i.i.d. random variables. Let i.i.d random variables X n, n N = {1, 2,... }, be defined on a probability space Ω, F, P). Denote S n = X 1 + +X n, V 2 n = X X 2 n, n N. A self-normalized sum S n /V n is correctly defined on the set {V n > 0}. For S n /V n to be defined on the entire set Ω, we put S n /V n = 0 on the set {V n = 0}. Define a function λn), n N, by putting λn) = 1 for n = 1,..., 9 and λn) = [n n/ ln ln n] for n N, n 10. Square bracets stand for an integer part function of the number in the bracets. The nonnegative integer function λn), n N, has the following properties: λn) λn + 1) for all n N and λn) < n for n N, n 10, λn) lim n = 1, lim n λn)) =. 1) Theorem 1.1. i) If a sequence {S n /V n } n 1 is wealy compact, then lim sup2 log log n) 1/2 S < a.e. 2) λn) n V ii) If random variables X n, n N, are symmetric, then lim sup2 log log n) 1/2 S 1 a.e. 3) λn) n V As an immediate consequence, we get statements from the article [2]. Corollary 1.2. i) If a sequence {S n /V n } n 1 is wealy compact, then lim sup S n V n 2 log log n < a.e. ii) If random variables X n, n N, are symmetric, then lim sup S n V n 2 log log n 1 a.e. 2 Supporting lemmas Virtually all of the following lemmas are nown to specialists. Let F n = σs, V 2, n) denote a σ-algebra generated by random variables S, V, n.

3 Igor Zhdanov 147 have Lemma 2.1. If E X 1 <, then for any n N and = 1,..., n + 1 we ES F n+1 ) = n + 1 S n+1 a.e. 4) Proof. Equalities similar to 4) are offered as an exercise in some advanced textboos on probability theory. It is also used in [2]. We were unable to find a source that has a proof of this equality. At the same time, the proof of our theorem is based on equality 4). For this reason we will give its proof here. It suffices to show the equality of conditional mathematical expectations EX F n+1 ) = EX 1 F n+1 ) a.e. for any = 1,..., n + 1. Then 4) follows from additivity of conditional mathematical expectation. Let L denote a class of events A F n+1, for which the following holds X dp = X 1 dp. 5) A Since X 1 X are i.i.d, equality 5) holds with A = Ω. If equality 5) holds for B, C F n+1 and B C, then it holds for A = C \ B due to integral s property of linearity. If equality 5) holds for events A n, n N, and A n A n+1 for all n N, then it holds for A = n=1a n by the dominated convergence theorem. This implies that class L is a λ-class. Let C denote a class of events such as A = {S 1 < c 1,,..., S r < c r, V 2 m 1 < b 1,..., V 2 m l < b l } for natural numbers r, l, n < < r, n + 1 m 1 < < m l and for real numbers c 1,..., c r, b 1,..., b l. It is easy to verify, using the integration measure change theorem, that equality 5) holds for any A C. Intersection of any two events from class C is in C. In other words, class C is a π-class. Clearly, π-class C generates σ-algebra F n+1. By Sierpinsi s theorem [4], p. 5), sigma-algebra σc) generated by class C is in λ-class L. Since F n+1 = σc) L F n+1, it follows that L = F n+1. Thus we proved that equality 5) holds for any A F n+1, which is equivalent to equality EX F n+1 ) = EX 1 F n+1 ) a.e. This completes the proof of Lemma 1. The following lemma demonstrates an important property of random variables Y = n n + m A S n+m V n+m ) 2, n, m N, = 1,..., m. Lemma 2.2. Random variables Y, = 1,..., m, form a submartingale relative to filtration F n+m, = 1,..., m.

4 148 Self-normalized laws of the iterated logarithm Proof. In essence, a proof of this lemma is provided in [2]. We include it here for the reader s convenience. It is easily seen that for any = 1,..., m the random variable Y is measurable with respect to sigma-algebra F n+m, and EY <. To complete the proof of the lemma, we have to verify that the submartingale condition Y EY +1 F n+m ) holds a.e. for any 1 < m. We will mae an additional assumption EX 2 1 <. By equality 4), an obvious inequality V n+m 1 V n+m as well as Jensen s inequality for conditional mathematical expectations ES n+m 1 F n+m )) 2 ES 2 n+m 1 F n+m ) a.e., we have n S ) 2 Fn+m ) n+m 1 EY +1 F n+m ) = E n + m 1 V n+m 1 n 1 ) 2ES 2 n + m 1 V n+m 1 F n+m ) n+m n 1 ) 2ESn+m 1 F n+m )) 2 n + m 1 V n+m n 1 ) 2 n + m 1 = n + m 1 V n+m n + m S n+m n S ) 2 n+m = = Y a.e. n + m V n+m We will now no longer assume that EX1 2 <. By A denote the indicator function of an event A F and X r) = X { X r} for = 1,..., n and r N. Define random variables S r) Y. It is not difficult to see that = X r) X r) and Y r) ) 2 similarly to S and lim Y r) = Y a.e., lim E Y r) Y = 0. 6) Applying the arguments shown above, we have Y r) EY r) +1 F n+m ) a.e. Letting r, we obtain inequality Y EY +1 F n+m ) a.e. It is well nown that conditions 6) are sufficient to justify the limit operation on conditional mathematical expectations. This completes the proof of Lemma 2. 3 Main Results i). Denote n r = [exp{r/log r) 2 ] for any r N, r 2 and n 1 = 1.

5 Igor Zhdanov 149 By Lemma 2 for n = λn r ) and m = n r+1 λn r ), n r > 10, random variables Y, = 1,..., m, form a submartingale relative to filtration F n+m, = 1,..., m. For any t > 0 a convex function exp{tx), x 0, is increasing. Therefore, a sequence exp{ty }, = 1,..., m, forms a submartingale relative to filtration F n+m, = 1,..., m. Denote { A r = λn r ) λn r) <n r+1 λn r ) + Event A r can be written as { A r = 1 m S V > x2 log r) 1/2 }, r N, n r > 10, x > 0. n S n+m } > x2 log r) 1/2 = { n + m V exp{ty } > exp{2x 2 t log r}} n+m 1 m By the imum inequality for submartingales [4], p. 93) we have P{A r } exp{ 2x 2 t log r}ee ty m = exp{ 2x 2 t log r}e exp{ts 2 λn r )/V 2 λn r )}. In [5] inequality sup n 1 E exp{cs 2 n/v 2 n } 2 is proven for some constant c > 0. Putting t = c and x = 2/c in 7), we obtain P{A r } 2/r 2, n r > 10, and P{A r } r=1 r:n r 10 By the Borel-Cantelli lemma, we have + r:n r>10 lim sup2 log r) 1/2 λn r ) S λn r ) <n r+1 λn r ) + V 2 r 2 <. x = 2 2/c a.e. It is not difficult to verify that lim n r+1 /n r = 1 and lim log log n r / log r = 1. This along with the first statement in 1) implies that lim λn r )/n r+1 = 1. From this, in turn, it follows that lim sup2 log log n r ) 1/2 S 2 2/c a.e. 8) λn r ) <n r+1 V For any n N there exists r N such that n r n < n r+1 and, consequently, the following inequality holds This together with 8) proves 2). S S. λn) n V λn r ) <n r+1 V 7)

6 150 Self-normalized laws of the iterated logarithm ii). If random variables X n, n N, are symmetric, then E exp{ts 2 n/v 2 n } 2/1 2t) for any t < 1/2. This statement is proven in the article [2]. Now inequality 7) can be restated as follows P{A r } exp{ 2x 2 t log r}ee ty m = exp{ 2x 2 2 t log r} 1 2t, t < 1 2, n r > 10. Put x = 1 + ε, where ε is any positive number. There exists t, 0 < t < 1/2, such that 21 + ε) 2 t > 1. For such x and t the previous inequality implies that P{A r } r=1 r:n r 10 By the Borel-Cantelli lemma we have PA r } t r:n r >10 lim sup2 log r) 1/2 λn r ) S λn r) <n r+1 λn r ) + V 1 r 2x2 t <. 1 + ε a.e. Properties of numbers n r, r N, as well as the function λn), n N, referred to in the proof of the first statement, imply that lim sup2 log log n r ) 1/2 S λn r ) <n r+1 V 1 + ε a.s. Using concluding arguments from the proof of the first statement, we can again verify that lim sup2 log log n) 1/2 S 1 + ε a.e. 9) λn) <n V This inequality holds on an event Ω ε of probability one. Inequality 9) holds on the event Ω = l=1 Ω 1/l of probability one for any ε = 1/l, l N. Letting ε = 1/l 0 in 9), we get 3). This completes the proof of the theorem.

7 Igor Zhdanov 151 References [1] N.M. Bingham, Variants of law of the iterated logarithm, Bull. London Math. Soc., 18, 1986), [2] E. Gine and D.M. Mason, On the LIL for self-normalized sums of iid random variables, J. Theor. Probab., 11, 1998), [3] Ph. Griffin, Kuelbs J.D. Self-normalized laws of the iterated logarithm, Ann. Probab., 17, 1989), [4] J. Yeh, Martingales and Stochastic Analysis, World Scientific, 1995.