On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables

On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables Deli Li 1, Yongcheng Qi, and Andrew Rosalsky 3 1 Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota, USA 3 Department of Statistics, University of Florida, Gainesville, Florida, USA Abstract: For a sequence of nondegenerate i.i.d. bounded random variables {Y n, n 1} defined on a probability space Ω, F, P) and a constant b > 1, it is shown for W n = b 1) bi Y i /b n+1 that S F Y1 ) {c R : P {ω Ω : c is a limit point of W n ω)}) = 1} = S F V ) [l, L] and that for almost every ω Ω, the set of limit points of W n ω) coincides with the set S F V ) where S F Y1 ) and S F V ) are the spectrums of the distribution functions of Y 1 and V b 1) b i Y i, respectively and l and L are the essential infimum of Y 1 and the essential supremum of Y 1, respectively. Examples are provided showing that, in general, the above two inclusions are proper. Keywords: Almost sure convergence; Essential infimum; Essential supremum; Infinite Bernoulli convolution; Geometric weights; Iterated logarithm type behavior; Law of the iterated logarithm; Limit points; Spectrum of a distribution function; Sums of geometrically weighted i.i.d. random variables. Mathematics Subject Classification: 60F15. Running Head: Limit Points of Normed Sums The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176. Address correspondence to Andrew Rosalsky, Department of Statistics, University of Florida, Gainesville, FL 3611-8545, USA; E-mail: rosalsky@stat.ufl.edu 1. INTRODUCTION At the origin of the current investigation is that of Rosalsky [16] concerning iterated logarithm type behavior for geometrically weighted independent and identically distributed i.i.d.) bounded random variables. The result of Rosalsky [16] pertaining to the current 1

work will now be discussed. We recall that the essential infimum and the essential supremum of a bounded random variable Y, denoted by ess inf Y and ess sup Y, respectively, are defined by ess inf Y = sup{y : PY y) = 1} and ess sup Y = inf{y : PY y) = 1}. Rosalsky [16] obtained the following result which describes the almost sure a.s.) asymptotic fluctuation behavior of the partial sums bi Y i where {Y n, n 1} is a sequence of nondegenerate i.i.d. bounded random variables defined on a probability space Ω, F, P) and b > 1 is a constant. The b i, i 1 are referred to as geometric weights and the partial sums bi Y i, n 1 are thus referred to as sums of geometrically weighted i.i.d. random variables. Theorem 1.1. Rosalsky [16]). Let {Y n, n 1} be a sequence of nondegenerate i.i.d. bounded random variables and let b > 1. Then lim sup bi Y i b n+1 /b 1) = L a.s. and lim inf where L = ess sup Y 1 and l = ess inf Y 1. bi Y i b n+1 /b 1) = l a.s. Theorem 1.1 sharpens a result of Teicher [1] Theorem 5) which asserts that the strong law of large numbers lim bi Y i = 0 a.s. b n c n holds for every numerical sequence c n no matter how slowly). Now one of the most striking and fundamental results of probability theory is the renowned law of the iterated logarithm due to Hartman and Wintner [10] for a sequence of i.i.d. square integrable random variables. This result, which has long been celebrated as being a crowning achievement in the theory of sums of independent random variables, is stated as follows. Theorem 1.. Hartman and Wintner [10]). Let {X n, n 1} be a sequence of i.i.d. random variables defined on a probability space Ω, F, P). Suppose that EX 1 = 0 and EX1 = 1. Then lim sup X i n log log n = 1 a.s. and lim inf X i n log log n = 1 a.s. Alternative proofs of the Hartman-Wintner theorem were discovered by Strassen [0], Heyde [11], Egorov [6], Teicher [1], Csörgő and Révész [4] p. 119), de Acosta [1], and Einmahl and Li [7].

We recall that for a given sequence {t n, n 1} in R, a point c [, ] said to be a limit point of {t n, n 1} if some subsequence of {t n, n 1} has c as its limit: lim t n k = c for some positive integer sequence {n k, k 1} with n k < n k+1, k 1. k It is convenient to simply say that c is a limit point of t n. Remark 1.1 i). It is well known see, e.g., Petrov [15] p. 48)) that the following sharpening of Theorem 1. holds: Under the hypotheses of Theorem 1., { { c [, ] : P ω Ω : c is a limit point of X }) } iω) = 1 n log log n 1.1) = [ 1, 1]. Hence for all c [ 1, 1], there exits an event Ω c with P Ω c ) = 1 such that for all ω Ω c, there exists a strictly increasing sequence of positive integers {n k, k 1} such that k X iω) = c. nk log log n k lim k The integer sequence {n k, k 1} is of course random; that is, the n k, k 1 depend on ω Ω c. ii). An even stronger version of the limit point result 1.1) indeed prevails; namely see, e.g., Stout [19] p. 94)), the set of limit points of X iω) n log log n coincides with the set [ 1, 1] for almost every ω Ω. 1.) The difference between 1.1) and 1.) is that for c [ 1, 1], 1.1) does not preclude the event of probability 1 { ω Ω : c is a limit point of X } iω) n log log n from depending on c whereas 1.) ensures that there exists an event Ω with P Ω ) = 1 such that for all c [ 1, 1] and all ω Ω, c is a limit point of X iω) ; n log log n that is, Ω does not depend on c [ 1, 1]. In the current work, for a sequence of nondegenerate i.i.d. bounded random variables {Y n, n 1} and a constant b > 1, we investigate which points c R are a.s. limit points of the sequence of normed weighted sums {W n, n 1} defined by W n = bi Y i b n+1 /b 1), n 1 1.3) 3

whose a.s. lim sup and a.s. lim inf were identified in Theorem 1.1. Let n V n = b 1) b i Y i, n 1. 1.4) Then and b 1) E ) b i Y i = b 1) EY1 ) b i = EY 1 b 1) Var ) b i Y i = b 1) Var Y 1 ) b i = b 1 b + 1 Var Y 1 and so by the Khintchine-Kolmogorov convergence theorem see, e.g., Chow and Teicher [3] p. 113)), V n converges a.s. to a random variable V 1.5) with EV = EY 1 and Var V = b 1 b + 1 Var Y 1. 1.6) A pertinent observation is that W n and V n are identically distributed, n 1. 1.7) For a random variable X with distribution function F X x) = PX x), x R, we let S F X ) denote the spectrum of F X ; that is S F X ) = {c R : F X c ε) < F X c + ε) for all ε > 0}. It is well known and easy to show that S F X ) is a closed set. In Theorem 3.1, it is shown inter alia that {c R : P {ω Ω : c is a limit point of W n ω)}) = 1} = S F V ) and in Theorem 3., it is shown that for almost every ω Ω, the set of limit points of W n ω) coincides with the set S F V ). Consequently, Theorem 3. bears the same relation to Theorem 3.1 for sums of geometrically weighted i.i.d. random variables as in Theorem 1.1 as does 1.) to 1.1) for sums of i.i.d. random variables as in Theorem 1.. The plan of the paper is as follows. In Section, we prove two preliminary lemmas. The main results of this paper, Theorems 3.1 and 3., are stated and proved in Section 3. In Section 4, two interesting examples are presented. 4

. PRELIMINARY LEMMAS To establish the main results, two lemmas are needed. The first lemma concerns a numerical sequence and is used to prove the second lemma which concerns an arbitrary sequence of random variables. Lemma.1. Let {t n, n 1}be a sequence in R and let c R. Then c is a limit point of {t n, n 1}.1) if and only if for all ε > 0, t n c < ε for infinitely many n..) Proof. Necessity. This implication is evident. Sufficiency. Suppose that.) holds. Choose n 1 1 such that t n1 c < 1 and n > n 1 such that t n c < 1. Suppose, inductively, that for some k, there exists integers n 1,..., n k with 1 n 1 < < n k such that Choose n k+1 > n k so that tnj c < 1 j for j = 1,..., k. t nk+1 c < 1 k + 1. Thus {n k, k 1} is a strictly increasing sequence of positive integers with t nk c < 1 k, k 1. Let ε > 0 and let K be such that K 1/ε. Then for all k K, and so lim k t nk = c; that is,.1) holds. t nk c < 1 k 1 K ε Lemma.. Let {T n, n 1} be a sequence of random variables and let c R. Then the following three statements are equivalent: i) c is a limit point of T n a.s. ii) For all ε > 0, P T n c ε, c + ε) i.o.n)) = 1. [ ) ]) iii) P j=1 T n c 1, c + 1 i.o.n) = 1. j j Proof. The equivalence between ii) and iii) is evident and the equivalence between i) and iii) follows immediately from Lemma.1. Remark.1. Stout [19] p. 33) defines a point c R to be a recurrent state of the sequence {T n, n 1} if ii) of Lemma. holds. 5

3. THE MAIN RESULTS With the preliminaries accounted for, the main results may be stated and proved. Throughout the rest of the paper, let {W n, n 1}, {V n, n 1}, and V be as in 1.3), 1.4), and 1.5), respectively, let F Y1 and F V denote the distribution functions of Y 1 and V, respectively, let C = {c R : P {ω Ω : c is a limit point of W n ω)}) = 1}, and let C ω) denote the set of limit points of W n ω), ω Ω. Theorem 3.1. Let {Y n, n 1} be a sequence of nondegenerate i.i.d. bounded random variables defined on a probability space Ω, F, P) and let b > 1. Then where l = ess inf Y 1 and L = ess sup Y 1. Proof. To prove S F Y1 ) C, let c S F Y1 ). Set p ε = F Y1 c + ε ) F Y1 c ε ) = P S F Y1 ) C = S F V ) [l, L] 3.1) Y 1 c ε, c + ε ]), ε > 0. Then p ε > 0 for all ε > 0 since c S F Y1 )) and p ε < 1 for all sufficiently small ε > 0 since Y 1 is nondegenerate). Let [ ] log k n k = k and j k = n k + 1, k 1 log p 1 ε where ε > 0 is small enough so that p ε < 1. Note that [ ] log k n k + 1 j k = as k. 3.) log p 1 ε Now for k 1, or, equivalently, Moreover, n k + 1 j k log k log p 1 ε = log k 1 log p ε p n k+1 j k ε k 1. 3.3) j k < n k < j k+1 for all large k. 3.4) 6

Then since the {Y n, n 1} are independent, P k=1 nk i=j k [ Y i c ε ]] ), c + ε = = k=1 k=1 F Y1 c + ε ) F Y1 c ε )) nk j k +1 p n k j k +1 ε k 1 by 3.3)) k=1 =. Moreover, the independence of the {Y n, n 1} and 3.4) ensure that for all large K, the events n k i=j k [ Y i c ε, c + ε ]], k K are independent and so by the Borel-Cantelli lemma, nk [ P Y i c ε, c + ε ]] i=j k i.o.k) ) = 1. 3.5) Next, note that j k 1 b i Y i j k 1 b i Y i j k 1 b i L l ) bj k L l ) b 1 and so jk 1 bi Y i b n k+1 /b 1) L l b n k+1 j k 0 a.s. as k by 3.)). 3.6) Moreover, it follows from 3.5) that with probability 1, for infinitely many k, k i=j k b i { Y nk i i=j k b ) i c ± ε b n k+1 /b 1) b n k+1 /b 1) = = b n k+1 b j k b n k+1 1 1 b n k+1 j k c ± ε ) ) c ± ε ) 3.7) { c ± 3 ε 7

provided k is sufficiently large by 3.)). Then by 3.6) and 3.7), with probability 1, for infinitely many k, jk 1 nk W nk = bi Y i b n k+1 /b 1) + i=j k b i Y i c ε, c + ε); b n k+1 /b 1) that is, and, a fortiori, Then by Lemma., thereby proving that S F Y1 ) C. Next, it will be shown that P W nk c ε, c + ε) i.o. k)) = 1 P W n c ε, c + ε) i.o. n)) = 1. P {ω Ω : c is a limit point of W n ω)}) = 1 3.8) S F V ) C. 3.9) Let c S F V ) and let ε > 0 be arbitrary. Let q ε = F V c + ε ) F V c ε ) = P c ε 4 4 4 < V c + ε ). 4 Then q ε > 0 since c S F V ). Note that V V n b 1) b i Y i i=n+1 Let Nε) be a positive integer such that Set Then for n > m 1, U m, n) = b 1 b n+1 L l b n a.s., n 1. 3.10) L l b Nε) < ε 4. 3.11) n i=m+1 U m, n) W n = b 1 b n+1 = b 1 b n+1 = b 1 b n m = W m b n m. 8 b i Y i, n > m 1. n i=m+1 b i Y i m ) b i Y i m ) bi Y i b m+1 ) n b i Y i 3.1)

Now for j 1, and, similarly, Thus W j j ) b 1) bi L b 1 b j+1 b j+1 W j l a.s. bj+1 b L L a.s. b 1 W j L l a.s., j 1. 3.13) Then by 3.1), 3.13), and 3.11), whenever m 1 and n m + Nε) we have Um, n) W n = W n L l bn m b < ε n m 4 and W m L l bn m b < ε n m 4 Now whenever m 1 and n m + Nε), 0 < q ε = P c ε 4 < V c + ε ) 4 a.s. 3.14) a.s. 3.15) = P c ε 4 < V c + ε 4, ε 4 < V n V < ε ) 4 by 3.10) and 3.11)) P c ε < V n < c + ε ) = P c ε < W n < c + ε ) by 1.7)) = P c ε < W n < c + ε, ε 4 < Um, n) W n < ε ) 4 by 3.14)) P c 34 ε < Um, n) < c + 34 ) ε. Let n k = knε), k 1. Then k=1 P c 34 ε < U n k, n k+1 ) < c + 34 ) ε = and since {U n k, n k+1 ), k 1} is a sequence of independent random variables, we have by the Borel-Cantelli lemma that P c 34 ε < U n k, n k+1 ) < c + 34 ) ε i.o.k) = 1. 3.16) 9

Now whenever m 1 and n m + Nε), we have up to an event of probability 0 that [c 34 ε < Um, n) < c + 34 ε ] = = [ c 3 4 ε < W n W m b < c + 3 ] n m 4 ε by 3.1)) [ c 3 4 ε < W n W m b < c + 3 n m 4 ε, ε 4 < W m b < ε ] n m 4 by 3.15)) [c ε < W n < c + ε]. Then recalling 3.16), 1 = P c 34 ε < U n k, n k+1 ) < c + 34 ) ε i.o.k) and, a fortiori, P c ε < W nk+1 < c + ε i.o.k) ) P c ε < W n < c + ε i.o.n)) = 1. Thus c C by Lemma. since ε > 0 is arbitrary thereby establishing 3.9). It will now be shown that C S F V ). 3.17) Suppose that c / S F V ). Then for some ε 0 > 0, Now recalling 3.10), we have for all large n that Then for all large n, P c ε 0 < W n < c + ε 0 ) F V c + ε 0 ) F V c ε 0 ) = 0. 3.18) = P c ε 0 < V n < c + ε 0 ) by 1.7)) V V n < ε 0 a.s. 3.19) = P c ε 0 < V n < c + ε 0, ε 0 < V V n < ε 0 ) by 3.19)) P c ε 0 < V < c + ε 0 ) F V c + ε 0 ) F V c ε 0 ) = 0 by 3.18)) 10

and so P c ε 0 < W n < c + ε 0 i.o.n)) = 0. Hence by Lemma., c / C. Consequently, 3.17) holds and so, recalling 3.9), C = S F V ). It remains to prove that C [l, L]. Let c C; that is, c R satisfies 3.8). Then by Theorem 1.1, l = lim inf W n c lim sup W n L a.s. n and so c [l, L] thereby proving that C [l, L]. Alternatively, it can be noted that V = b 1) b i Y i [l, L] a.s. and so C = S F V ) [l, L].) Remark 3.1 i). Recalling 1.5) and 1.6), and recalling Theorem 1.1, V n V a.s. and V is nondegenerate 3.0) lim inf W n = l a.s. and lim sup W n = L a.s. 3.1) Thus in view of 3.0) and 3.1), we see that {V n, n 1} and {W n, n 1} exhibit entirely different asymptotic behavior although their marginal distributions coincide for all n 1 as was noted in 1.7). ii). It follows from 3.0) that V n d V and hence by 1.7) W n d V. 3.) It is interesting to contrast 3.1) and 3.) with what prevails for the sequence H n X n, n 3 n log log n in the Hartman-Wintner Theorem Theorem 1. above) where {X n, n 1} is a sequence of i.i.d. random variables with EX 1 = 0 and EX 1 = 1. Recall that Theorem 1. asserts that lim inf H n = 1 a.s. and lim sup H n = 1 a.s. 3.3) Using Chebyshev s inequality, H n P 0 which is equivalent to H n d 0. 3.4) While 3.1) and 3.3) are analogous results in terms of their structure, 3.) and 3.4) are structurally different since V is nondegenerate whereas the random variable 0 is degenerate. iii). We also observe that there does not exist a random variable W such that W P n W. For if such a random variable W exists, then W d n W whence by 3.) necessarily W and V are identically distributed and, moreover, there exists a nonrandom) subsequence {n k, k 1} such that W nk W a.s. But by the Kolmogorov 0 1 law, W = lim sup k W nk is degenerate which contradicts V being nondegenerate. 11

In the next theorem, we show that for almost every ω Ω, the random) set C ω) equals the nonrandom) set S F V ) = C by Theorem 3.1). This is a strengthening of the assertion c S F V ) if and only if c R and with probability 1, c is a limit point of W n obtained from Theorem 3.1. Theorem 3.1 is used to prove Theorem 3.. Theorem 3.. Let {Y n, n 1} be a sequence of nondegenerate i.i.d. bounded random variables defined on a probability space Ω, F, P) and let b > 1. Then the set C ω) coincides with the set S F V ) for almost every ω Ω. 3.5) Proof. We first prove that S F V ) C ω) for almost every ω Ω. 3.6) For each x S F V ), by Theorem 3.1 there exists an event Ω x with P Ω x ) = 1 such that for all ω Ω x, x is a limit point of W n ω). Since S F V ) is closed and since S F V ) [l, L] by Theorem 3.1, S F V ) is compact. Thus see, e.g., Royden [17] p. 163)) S F V ) is separable; that is, S F V ) contains a countable set K which is dense in S F V ). Set Ω 0) = x K Ω x. Since K is countable, Ω 0) is an event with P Ω 0)) = 1. Now for every ω Ω 0) and every x K, x is a limit point of W n ω). 3.7) Fix ω 0 Ω 0) and x 0 S F V ). Let ε > 0. Since K is dense in S F V ), there exists x 1 K such that x 0 x 1 < ε/. Now by 3.7) and Lemma.1, W n ω 0 ) x 1 < ε for infinitely many n. Then W n ω 0 ) x 0 W n ω 0 ) x 1 + x 1 x 0 < ε + ε = ε for infinitely many n and so by Lemma.1, x 0 C ω0 ). Since ω 0 P Ω 0)) = 1, the assertion 3.6) holds. Ω 0) and x 0 S F V ) are arbitrary and Next, it will be shown that Set Ω 1) = C ω) S F V ) for almost every ω Ω. 3.8) { ω Ω : lim inf Then P Ω 1)) = 1 by Theorem 1.1 and W nω) = l and lim sup } W n ω) = L. Ω 1) { ω Ω : C ω) [l, L] }. 3.9) 1

Now S F V ) [l, L] by Theorem 3.1. If S F V ) = [l, L], then by 3.9) Ω 1) { ω Ω : C ω) S F V ) } and so 3.8) holds since P Ω 1)) = 1. Thus it will be assumed that S F V ) is a proper subset of [l, L]. Now by Theorem 3.1, {l, L} S F Y1 ) S F V ) [l, L] and since S F V ) is a closed set, it follows that [l, L] S F V ) = [l, L] S F V )) c = l, L) S F V )) c is an open set and hence see, e.g., Royden [17] p. 39)) can be expressed as [l, L] S F V ) = a i, b i ) 3.30) for some collection {a i, b i ), i 1} of nonempty) open intervals. Suppose we can show that there exists an event Ω ) Ω 1) with P Ω )) = 1 such that Then [l, L] S F V ) [l, L] C ω), ω Ω ). 3.31) C ω) S F V ), ω Ω ) which establishes 3.8). We now verify 3.31) for some event Ω ) Ω 1) with P Ω )) = 1. Note that for all k 1, it follows from 3.10) that there exists an integer N k 1 such that Then for all i 1, k 1, and n N k, P V n a i + 1 k, b i 1 )) k V V n 1 k a.s. for all n N k. 3.3) = P V n a i + 1 k, b i 1 ), V V n 1 ) k k by 3.3)) P a i < V < b i ) = 0 since a i, b i ) [l, L] S F V )) and hence by 1.7), for all i 1, k 1, and n N k P W n a i + 1 k, b i 1 )) k = 0. 3.33) 13

Set Ω ) = k=1 Ω 1) n=n k [ W n / a i + 1 k, b i 1 )] ). k Then Ω ) Ω 1) and it follows from 3.33) that P Ω )) = 1. Now Ω ) = = k=1 k=1 { Ω 1) ω Ω : C ω) [l, L] a i + 1 k, b i 1 )}) k { Ω 1) ω Ω : a i + 1 k, b i 1 ) }) [l, L] C ω) k Ω 1) { }) ω Ω : a i, b i ) [l, L] C ω) = Ω 1) { ω Ω : [l, L] S F V ) [l, L] C ω) } by 3.30)) thereby proving 3.31) and hence 3.8) as was noted above. Combining 3.6) and 3.8) yields the conclusion 3.5). The following corollary is an immediate consequence of Theorems 3.1 and 3.. Corollary 3.1. Let {Y n, n 1} be a sequence of nondegenerate i.i.d. bounded random variables defined on a probability space Ω, F, P) and let b > 1. If S F Y1 ) = [l, L] where l = ess inf Y 1 and L = ess sup Y 1, then C ω) = C = [l, L] for almost every ω Ω. 4. SOME INTERESTING EXAMPLES In this section we present two interesting examples pertaining to Theorem 3.1. The first example shows that, in general, the first inclusion in 3.1) of Theorem 3.1 is proper. In this example, S F Y1 ) = { 1, 1} but C = [ 1, 1]. Example 4.1. Let {Y n, n 1} be a sequence of i.i.d. random variables with P Y 1 = 1) = P Y 1 = 1) = 1. Then S F Y 1 ) = { 1, 1}. In Theorem 3.1, take b =. Now V n = n i Y i converges a.s. to a random variable V. It is well known that V is uniformly distributed on [ 1, 1] see, e.g., Chow and Teicher [3] Exercise 8.3.7, p. 93)) for a routine characteristic function argument). Thus C = [ 1, 1] by Theorem 3.1. The earliest reference we can find to V having a uniform distribution is Jessen and Wintner [1]. 14

Remark 4.1. One of the great problems of twentieth century probability theory which has not yet been completely solved is that of identifying the type of the distribution function of V = b i Y i where {Y n, n 1} is a sequence of i.i.d. random variables with P Y 1 = 1) = P Y 1 = 1) = 1 and b > 1. The distribution function F V is known as an infinite Bernoulli convolution. There is an interesting literature of investigation on this problem going back to the groundbreaking work of Jessen and Wintner [1] and Erdös [8]. Jessen and Wintner [1] showed that F V cannot be discrete and it cannot be a mixture; it must be either continuous singular or absolutely continuous. Erdös [8] showed that F V is continuous singular if b >. A review of earlier developments concerning this problem is contained in Garsia [9]. Garsia [9] also pointed out several references of V and F V arising in connection with psychological experimentation or in data transmission problems. We refer the reader to the more recent work of Diaconis and Freedman [5] for a concise discussion of this problem of identifying the type of F V ; this work contains a brief discussion of the substantial recent discovery of Solomyak [18] whose proof was subsequently simplified by Peres and Solomyak [13]) asserting that F V is absolutely continuous for almost every [Lebesgue measure] value of b 1, ]. Solomyak s [18] result had previously been conjectured by Garsia [9].) The exceptional set is nonempty. For example, Erdös [8] showed for b = 1+ 5)/ = the golden ratio ϕ) that F V is continuous singular. The problem of characterizing the type of the distribution function F V of V = b i Y i where {Y n, n 1} is a sequence of nondegenerate i.i.d. random variables with P Y 1 = 1) = 1 P Y 1 = 1) 1 was investigated by Peres and Solomyak [14]. The second example shows that, in general, the second inclusion in 3.1) of Theorem 3.1 is also proper; we exhibit a point c l, L) such that P {ω Ω : c is a limit point of W n ω)}) = 0. Example 4.. Let {Y n, n 1} be a sequence of i.i.d. random variables with P Y 1 = 0) = P Y 1 = 1) = 1. Then l = ess inf Y 1 = 0 and L = ess sup Y 1 = 1. Let b >. Note that for almost every ω [Y n = 0] where n W n ω) = 1 bi Y i ω) b n+1 /b 1) b 1) n 1 bi b n+1 and that for almost every ω [Y n = 1] where n Thus for all n, Now since b >, and so letting c W n ω) P W n < 1 b b 1)bn = b 1. b n+1 b = bn b b n+1 < 1 b or W n b 1 ) = 1. 4.1) b 1 b < b 1 b 1 b, b 1 ) l, L) and ε 0, c 1 ) )) b 1 c, b b b 15

it follows from 4.1) that for all n P W n c ε, c + ε)) = 0 and hence It follows that and so P W n c ε, c + ε) i.o.n)) = 0. lim inf W n c ε a.s. P {ω Ω : c is a limit point of W n ω)}) = 0. Remark 4.. In Example 4., if b = 3, then V n = 3 i Y i V a.s. where F V is the distribution function of the Cantor singular distribution see, e.g., Billingsley [] p. 416). Thus C = S F V ) is the Cantor ternary set. REFERENCES 1. de Acosta, A. 1983. A new proof of the Hartman-Wintner law of the iterated logarithm. Annals of Probability 11:70-76.. Billingsley, P. 1995. Probability and Measure, 3rd ed. John Wiley, New York. 3. Chow, Y.S., and Teicher, H. 1997. Probability Theory: Independence, Interchangeability, Martingales, 3rd ed. Springer-Verlag, New York. 4. Csörgő, M., and Révész, P. 1981. Strong Approximations in Probability and Statistics. Academic Press, New York. 5. Diaconis, P., and Freedman, D. 1999. Iterated random functions. SIAM Review 41:45-76. 6. Egorov, V. A. 1971. A generalization of the Hartman-Wintner theorem on the law of the iterated logarithm. Vestnik Leningradskogo Universiteta No. 7:-8 [in Russian; English translation in Vestnik Leningrad University. Mathematics 4:117-14]. 7. Einmahl, U., and Li, D. 005. Some results on two-sided LIL behavior. Annals of Probability 33:1601-164. 8. Erdös, P. 1939. On a family of symmetric Bernoulli convolutions. American Journal of Mathematics 61:974-976. 9. Garsia, A. M. 196. Arithmetic properties of Bernoulli convolutions. Transactions of the American Mathematical Society 10:409-43. 10. Hartman, P., and Wintner, A. 1941. On the law of the iterated logarithm. American Journal of Mathematics 63:169-176. 16

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