ROCKY MOUNTAIN JOURNA OF MATHEMATICS Volume 26, Number 1, Winter 1996 ON RANDOM WAKS AND EVES OF THE FORM na SARAH DRUCKER, IMOA FODOR AND EVAN FISHER 1. Introduction. et {Xi, i = 1, 2, 3,... } be a collection of independent and identically distributed random variables with finite expectation J and finite variance a 2 Define Sn = ~=l xi for n = 1, 2, 3,.... It follows from results such as the Hartman-Wintner law of the iterated logarithm (see [10, p. 293]) that if a > 1/2, then limn--+oo ( Sn - nj) / n a = 0 a.s. (The case a = 1 is the standard Strong aw of arge Numbers.) An equivalent statement is that for each number c: > 0 the inequality is satisfied for only finitely many n. Consider the sequence of events {An, n = 1, 2, 3,... } where An = (IBn- nji > en a). et I(A) denote the indicator function of the event A. We consider the random variable N(c:) where N(c:) is defined by (1.1) N(c:) =!(An) n=l N(c:) represents the number of indices for which An occurs. For notational convenience we suppress, in the notation, the dependence of N(c:) on a. For a > 1/2, this is a finite-valued random variable. Conditions related to the existence of various moments of N(c:) for a variety of boundaries and settings have been investigated by ai and an [5], Slivka and Severo [9], Slivka [8], Stratton [11], and Griffiths and Wright [4]. In Section 2 of this paper we consider the expected value of N(c:) for the special case when Xi is normally distributed. In Slivka and Received by the editors on November 9, 1993, and in revised form on August 17, 1994. The first and second authors were participants in NSF-REU program held at afayette College, Summer 1993. Copyright @1996 Rocky Mountain Mathematics Consortium 117
118 S. DRUCKER, I. FODOR AND E. FISHER Severo [9, Theorem 2], explicit upper and lower bounds are derived in this setting where a= 1. Using a simpler method we derive analogous upper and lower bounds for E(N(e)) for the more general case a> 1/2 (Theorem 2.3), which includes the aforementioned bounds as a special case. Theorem 2.4 describes the asymptotic behavior of E(N(e)) as e -t 0. This generalizes a result indicated in Slivka and Severo [9] for the case a = 1. In part 3 we consider the number of times the random walk { Sn, n = 1, 2, 3,... } lies above a linear boundary. More precisely, we define the random variable N + (e) as (1.2) N+(e) = I(Bn) n=l where Bn is the event defined by Bn = (Sn > n(j +e)) and I( ) is the indicator function defined earlier. This variable has been investigated in Razanadrakoto and Severo [6] where expressions generating the exact distribution of N+(e) are derived. Section 3.1 begins with the derivation of close upper and lower bounds for the expected value of N+(e) in the special case when Xi is normally distributed (Corollary 3.1). We also note an analogous result under the same conditions on the variance of N+(e) (inequality (3.3)). The primary results in Section 3 concern the first index, if one exists, for which (Sn > n(j +e)) occurs. We define the variable T(e) by (1.3) T(e) = { inf{n 2 1: Sn > n(j +e)} if such _ann exists oo otherwise. Under the hypothesis that {Xi, i = 1, 2, 3,... } are independent and identically distributed random variables with finite expectation, we derive a general recursive expression generating the exact distribution of T(e) (Theorem 3.3). The final results in Section 3 center around the conditional expectation of T(e) given that T(e) is finite. Theorem 3.4 describes a relationship between this conditional expectation and the expected value of N+(e). We then investigate the asymptotic behavior of the conditional expectation as e -t 0 in the case where the common distribution function of {Xi, i = 1, 2, 3,... } is symmetric and continuous (Theorem
RANDOM WAKS AND EVES OF THE FORM n"' 119 3.5) and in the special case where the common distribution function of {Xi, i = 1, 2, 3,... } is normal (Theorem 3.6). 2. The expected value of N(c). The following lemmas are employed at several points in the development. emma 2.1. If Y is a nonnegative random variable, then n=l n=l Proof. See Ash [1, p. 275]. D emma 2.2. If the random variable Z has a standard normal distribution, then for p > -1, EIZIP = 2P/2 r(p+ 1) ft 2 Proof. The result follows immediately using an elementary change of variables. D Theorem 2.3. et {Xi, i = 1, 2, 3,... } be a sequence of independent random variables, normally distributed with mean 1-l and variance a 2 et a> 1/2 and N(c) be as defined in (1.1). For c > 0, (2.1) p-2/(2a-1) 21/(2a-l)f((2a+1)/(2(2a-1))) - 1 ft where p = cfa. ::; EN(c) < _2;(2a-l) 21/(2a-l)f( (2a+ 1) / (2(2a-1))) _p ft
120 S. DRUCKER, I. FODOR AND E. FISHER Proof It follows from the definition of N(c) that EN( c)= P(ISn- nji > w<>) n=l n=l n=l where Z has a standard normal distribution. The application of emmas 2.1 and 2.2 to the latter sum yields inequality (2.1). D Remarks. 1) We note that, as expected, the bounds on E(N(c)) are based on c and a through p = c j a. 2) For the case a= 1, Theorem 2.3 reduces to Theorem 2 in Slivka and Severo [9]. Examples. The following examples demonstrate the relative closeness of the upper and lower bounds in Theorem 2.3 when p is small. 1) Consider the case a = 1 and a = 1. If c =.5 Theorem 2.3 yields the inequality 3 :S E(N(c)) :S 4. For c =.1, the inequality 99 :S E(N(c)) :S 1 is obtained from Theorem 2.3. 2) Consider the case a = 1 and a = 3/4. For c = 1, Theorem 2.3 yields 2 :S E(N(c)) :S 3. For c =.5, it follows that 47 :S E(N(c)) :S 48. For c =.1, we obtain the estimate that 29,999 :S E(N(c)) :S 30,0. Theorem 2.4 describes an asymptotic approximation for E(N(c)) as p -+ 0 for the general case a > 1/2. This includes as a special case a result noted in Slivka and Severo [9, p. 732] for a = 1. We apply a method of proof suggested in their paper. Theorem 2.4. Assume the same hypotheses as in Theorem 2.3. Then (2.2) lim IE(N(c)) _ {P- 2;c2a--1) 2 1 /( 2 a-- 1 lr((2a + 1)/2(2a- 1))) _~}I =O p-+0 Vir 2
RANDOM WAKS AND EVES OF THE FORM n"' 121 where p =c/o". Proof. et <I>( ) denote the cumulative distribution function for the standard normal distribution. Then, as in the proof of Theorem 2.3, we obtain E(N(c)) = 2 <I>( -pna-1 1 2). n=1 Define the function f by f(x) = <I>( -pxa-1 1 2) and the function ~ by ~(x) = J;([t]- t + 1/2) dt for x ~ 0. We apply a standard form of the Euler-Macaurin summation formula to the latter sum and obtain E(N(c)) = 2 roo f(x) dx + f(1) + lim f(n) + 2 roo J"(x)~(x) dx. 11 n--+oo 11 Since ~(x) :::; 1/8 for x > 0 and limn--+oo f(n) = 0, it follows that (2.3) IE(N(c))- 21 f(x) dx + 21 1 f(x) dx- <I>( -p)l :::; ~ 1 lf"(x)l dx. We apply the change of variables y = pxa-1 1 2 and integration by parts to the integral J f(x) dx with the result (2.4) 1oo f(x) dx = 1oo <I>( -pxa-1/2) dx = _2 -p-2/(2a-1) roo [1- <I>(y)]y(3-2a)/(2a-1) dy 2a-1 1o = p-2/(2a-1) { lim [1 - <P(y)]y2/(2a-1) y--+oo + 1 y2/(2a-1) (y) dy} where ( ) denotes the density function of the standard normal distribution. Using standard bounds on the tail probabilities of the standard normal distribution (see Feller [2, p. 175]) or other methods, it is easily shown that lim [1 - <I>(y )]y2/(2a-1) = 0. y--+oo
122 S. DRUCKER, I. FODOR AND E. FISHER Applying emma 2.2 with p = 2/(2a- 1) to the last integral in (2.4) results in (2.5) 1oo f(x) dx = P-2/(2a:-1)2(2-2a:)/(2a:-1)7r-1/2r((2a+1)/(2(2a-1))). Since.P( ) is bounded, the ebesgue dominated convergence theorem can be used to obtain (2.6) 11 11 1 lim.p( -pxa:-112) dx = lim.p( -pxa:-112) dx = -. p-+0 0 0 p-+0 2 We now show that (2.7) lim roo lf"(x)l dx = 0. p-+0 }1 Elementary calculations show that there exist constants C1 > 0 and c2 > 0 depending only upon a so that (2.8) If" (x)l :::; C1pxa:-5/2 exp (- p2x:a:-1 )+C2p3x3a:-7/2 exp (- p2x:a:-1). Consider Jt xa:-512 exp( -p2x2a:-1 /2) dx. In the case a < 3/2, it is clear that J 1 xa:-512 exp( -p2x2a:-1 /2) dx < C for some positive constant C independent of p. Therefore, for a < 3/2, it follows that /, oo ( p2x2a:-1) (2.9) limp xa:-512 exp - dx = 0. p-+0 1 2 Consider the case a > 3/2. The change of variables y = p2x2a:-1 /2 yields the existence of a constant C independent of p so that 1oo xa:-5/2 exp( -p2x2a:-1 /2) dx = C p(3-2a:)/(2a:-1) 1 y-(2a:-1)/2(2a:-1) exp( -y) dy. It follows from this that (2.9) holds for all a > 3/2.
RANDOM WAKS AND EVES OF THE FORM n"' 123 For a= 3/2, the same change of variables results in the equality (2.10) /, xa-5/2 exp( -p2x2a-1/2) dx = -11 y-1 exp( -y) dy. 1 2 p2 /2 The latter can be written as (2.11) r 1 y-1 exp( -y) dy + roo y-1 exp( -y) dy. 1p2j2 11 Clearly, limp--+o(1/2)p f 1 y- 1 exp( -y) dy = 0. We also observe that (2.12) r 1 y- 1 exp(-y)dy< r 1 y-1dy=ln2-2lnp. 1 p2 /2 1 p2 /2 Since limp--+o p ln p = 0, (2.10)-(2.12) yield the validity of (2.9) for the case a = 3/2 and hence for all a > 1/2. We now consider the integral J 1 x 301-7 12 exp( -p2x 201-1 /2) dx. For 1/2 < a < 5/6, it is clear that there exists a constant C independent of p so that J 1 x 301-7 12 exp( -p2x 201-1 /2) dx < C. Therefore, (2.13) lim p3 roo x301-712 exp( -lx201-1 /2) dx = 0. p--+0 11 The case a > 5/6 follows as before. With the change of variables y = p2x 201-1 /2, we obtain (2.14) loo x3a-7/2 exp( -p2x2a-1 /2) dx < 1oo x3a-7/2 exp( -p2x2a-1 /2) dx = c p(5-6a)/(2a-1) 1 y(2a-3)/(2(2a-1)) exp( -y) dy for some constant C independent of p. It is easily seen that the last integral converges. Therefore, (2.14) is valid for a > 5/6. Consider the case a= 5/6. We obtain (2.15)
124 S. DRUCKER, I. FODOR AND E. FISHER Write the latter integral as in (2.11). Clearly, (2.16) lim p3 rx; y-1 exp( -y) dy = 0. p-+o 11 From (2.14)-(2.16) and the fact that limp-+o p 3 lnp = 0, it follows that (2.13) holds for a= 5/6 and hence for all a > 1/2. Thus, (2.7) holds for all a > 1/2. Combining (2.5)-(2.7) proves (2.2). D Examples. 1) In the case a= 1, Theorem 2.4 demonstrates that lim IE(N(e))- (p- 2-1/2)1 = 0. p-+0 This is the same result described in Slivka and Severo [9, p. 732]. (Note there is a misprint in that paper.) 2) In the case a= 3/4, it follows from Theorem 2.4 that lim IE(N(e))- (3p- 4-1/2)1 = 0. p-+0 3. The variable N+(e). Recall the definition of N+(e) as given in (1.2). This variable was studied in Razanadrakoto and Severo [6] where, among other things, they derived an expression for the probability generating function of N + (e). 3.1. The expected value and variance of N+(e). We begin this section with two results that complement those in Section 2 of the present paper for the special case a = 1. Corollary 3.1. et {Xi,i = 1,2,3,... } be a sequence of i.i.d. random variables, normally distributed with mean f. and variance f5 2. Fore> 0, (3.1) where p = ejfj.
RANDOM WAKS AND EVES OF THE FORM n"' 125 Proof From the definition of N+(c-) in (1.2) we note that (3.2) n=l The result (3.1) follows from Theorem 2.3 and the symmetry of the random variables {Xi, i = 1, 2, 3,... }. D With the hypotheses and notation of Theorem 3.1, the following result on the variance of N + (c) holds: (3.3) 3p- 4 j4-1/8 :S Var[N+(c-)] :S 3p- 4 /4+ 1/8. The proof, which we omit, makes use of a series representation for the variance of N+(c-) derived in Razanadrokoto and Severo [6, p. 181] and follows along the lines of the proof of Theorem 2.4. 3.2. On the distribution and conditional expectation of the random variable T( c). et {Xi, i = 1, 2, 3,... } be a collection of independent and identically distributed random variables with finite expectation f. For Theorem 3.3 and Theorem 3.4 we make no further assumptions concerning the distribution of the random variables. Recall that the random variable T(c-), defined by (1.3), describes the first index, if one exists, for which the event (Sn > n(f. +c-)) occurs. It follows from Razanadrakoto and Severo [6] that T(c-) is a defective random variable, i.e., P(T(c-) < oo) < 1. As in their work, we consider the equivalent event that (S~ > 0) for S~ = ~=l x: with x: =xi- (f.+c-). For arbitrary c ~ 0, define Tn and an for n = 1, 2, 3,..., by (3.4) Tn = P(T(c-) = n) and (3.5) et T ( s) represent the generating function of { T n}. That is, we define T(s) by (3.6) T(s) = TnSn. n=l
126 S. DRUCKER, I. FODOR AND E. FISHER Applying a result in Feller [3, Theorem 1, p. 413], it is immediate that 1 oo n log 1 -r(s) = ~: P(S~ > 0) (3.7) oo n = ~P(Sn > n(f.+c)) n n=1 _~ann -.J -s. n n=1 In order to determine the distribution of T(c) from its generating function, we use the following lemma which is used for an analogous purpose in Razanadrokoto and Severo [6]. emma 3.2. et { ui, i = 0, 1, 2,... } and {vi, i = 0, 1, 2,... } be sequences of real numbers, and assume that U(s) = 2:;: 0 Ujsi and V(s) = 2:;: 0 Vjsi are such that U(s), V(s), and their nth derivatives exist for 0 < s < TJ for some TJ > 0. If U(s) = exp(v(s)) for 0 < s < TJ, then n -1~. Un = n.j)vjun-j, n = 1,2,... j=1 Proof. See Razanadrakoto and Severo [6, p. 180]. D Theorem 3.3 provides a recursive formula that generates the exact distribution of the defective random variable T(c). Theorem 3.3. et {Xi, i = 1, 2, 3,... } be independent and identically distributed random variables with finite expectation f.l et c ~ 0. Then, with the notation defined in (3.4) and (3.5), n (3.8) -1~ Tn = -n.j ajtn-j for n = 1, 2,..., j=1
RANDOM WAKS AND EVES OF THE FORM n"' 127 where To is defined as To = -1. Proof. With To as defined in the hypotheses, the relation described in (3.7) can be rewritten as (3.9) The result (3.8) is obtained by applying emma 3.2 with Un = -Tn and Vn = -anfn. D Example. et {Xi, i = 1, 2, 3,... } be a sequence of independent standard normal random variables. The following probabilities are obtained (correct to the number of decimal places shown) from Theorem 3.3: n P(T(1) = n) P(T(.5) = n) P(T(.01) = n) 1.158655.308538.499601 2.0267391.0722773.124917 3.830393.032322.0624605 4.318112.0179333.0390383 5.135864.0111152.027327 6.0620403.737263.0204954 7.0296445.511974.0161035 8.0146376.367504.0130842 9.7 40955.270495.0109035 10.382454.20304.926796 We have noted earlier that the variable T(c) is a defective random variable. Therefore, it is of interest to consider the conditional expectation of T(c) given that T(c) is finite. Theorem 3.4. et {Xi, i = 1, 2, 3,... } be independent and identically distributed random variables with finite expectation 1-l et c > 0.
128 S. DRUCKER, I. FODOR AND E. FISHER Then (3.10) Note. The probabilities appearing in (3.10) can be expressed as P(N+(e) = 0) = P(T(e) = oo) and P(N+(e) > 0) = P(T(e) < oo). Proof Since P(T(e) = nit(e) < oo) = P(T(~n < oo), it follows that (3.11) E[T(e)IT(e) < oo] = (~ Tn) - 1 ~ ntn et r(s) represent the generating function of {rn,n = 1,2,3,... } as defined in (3.6). Then (3.11) can be rewritten as (3.12) r'(1) E[T(e)IT(e) < oo] = 7 ( 1 ). A simple calculation using (3.7) leads to r'(s) ~ n-1 (3.13) 1 _ r(s) = ~ P(Sn > n(p, + e))s. Recall that (3.14) E(N+(e)) = P(Sn > n(p, +e)). n=1 Therefore, (3.13) and (3.14) imply that (3.15) r 1 (1) = (1- r(1))en+(e).
RANDOM WAKS AND EVES OF THE FORM n"' 129 We apply (3.15), (3.12), and the note following the statement of Theorem 3.4 to obtain E[T(c)IT(c) < oo] = 1- r(1) r(1) EN+(c) = P(T(c) = oo) EN (c) P(T(c) < oo) + = P(N+(c) = 0) EN ( ). P(N+(c) > 0) + c D Example. For {X,i = 1,2,3,... }, standard normal random variables, Corollary 3.1 provides close upper and lower bounds on E(N+(c)). Estimates for P(N+(c) = 0) can be obtained from the work of Razanadrakoto and Severo [6]. Theorem 3.4 can then be applied to obtain upper and lower bounds for E[T(c)IT(c) < oo]. For example, with c =.5, Theorem 3.4 establishes that 1.68::; E[T(.5)IT(.5) < oo]::; 2.25. We consider the behavior of E[T(c)IT(c) < oo] as c --+ 0 where the common distribution function of the random variables {Xi, i = 1, 2, 3,... } is continuous and symmetric. Theorem 3.5. et {Xi, i = 1, 2, 3,... } be independent and identically distributed random variables with common mean J and distribution function which is continuous and symmetric. Then, for c > 0, lim E[T(c)IT(c) < oo] = oo. e:-+0 Proof In order to emphasize the dependence of the probabilities an and Tn on c, denote an as an(c) and Tn as Tn(c). Then for each fixed index n, n = 1, 2, 3,..., (3.16) lim an(c) = 1/2 e:-+0 and (3.17) lim Tn(c) = T~ e:-+0
130 S. DRUCKER, I. FODOR AND E. FISHER where { T~, n = 1, 2, 3,... } is the solution to the recursive relation described in (3.8) with ai equal to 1/2. Due to the symmetry and continuity of the common distribution of the random variables {Xi,i = 1,2,3,... }, T~ can be equivalently characterized by T~ = P(Si ::; 0, s~ ::; 0,... 's~-1 ::; 0, s~ > 0) where s~ = 2::~=1 x: with x: = xi - f. As a result, the probability generating function for {T~,n = 1,2,3,... }, denoted here by T*(s), satisfies relation (3.7) (replace T with T*) with ai equal to 1/2. It follows from this that (3.18) Therefore, T*(s) = 1- v"f=s". nt~ = limt*'(s) = oo. n=1 stl Theorem 3.5 follows from this and (3.12). D For the case where {Xi, i = 1, 2, 3,... } are normally distributed random variables, we derive a result characterizing the asymptotic behavior of E[T(c-)IT(c-) < oo] as c- approaches 0. Specifically, we show in Theorem 3.6 that E[T(c-)IT(c-) < oo] increases to oo asymptotically as p-1 where p = c-ja. Theorem 3.6. et {Xi, i = 1, 2, 3,... } be independent and identically distributed random variables, normally distributed with mean f. and variance a 2 Then, for c > 0, (3.19) 0 < liminfpe[t(c-)it(c-) < oo]::; limsuppe[t(c-)it(c-) < oo] < oo p~o p~o where p = c-ja. Proof Define Pn for n = 1, 2, 3,..., by Pn = P(Sk > k(f. +c-), k = 1, 2, 3,..., n), and define Po by Po= 1. Razanadrokoto and Severo [6, p. 180] have shown that
RANDOM WAKS AND EVES OF THE FORM n"' 131 Combining this result with Theorem 3.4 leads to (3.20) E[T(c)IT(c) < oo] = ( ~Pk) - 1 EN+(c). We derive (3.19) by obtaining upper and lower bounds on the term 2::%"= 1 Pk and applying Corollary 3.1. In Razanadrokoto and Severo [6, p. 181] it is shown that (3.21) where ak is as defined in (3.5). et <I>( ) denote the cumulative distribution function for the standard normal distribution, and let ( ) denote its density function. From the monotonicity of the function f(x) = x<i>( -pft) for x > 0, we obtain (3.22) k=1 k=1 <<I>( -p) + 1 x- 1 <1>( -pft) dx. It is easily seen that 1 x- 1 <1>(-pft) dx = -2<1>(-p) lnp + 2l (lny) (y) dy (3.23) < -2<I>(-p)lnp+21 ycp(y)dy = -2<!>( -p) lnp + C where C is a positive constant independent of p. Applying the result described in (3.23) to (3.21), it follows that for p < 1 (3.24) :~:::>k < -1 + Cp- 2 ip(-p) < -1 + Cp- 1 k=1 for some positive constant C independent of p.
132 S. DRUCKER, I. FODOR AND E. FISHER The result in (3.23) can be combined with Corollary 3.1, and the result described in (3.20) to obtain the following inequality: For p < 1, there exists a positive constant C independent of p so that From this inequality, it follows that (3.25) 0 < liminf pe[t(c)it(c) < oo]. p-+0 We obtain the righthand inequality in (3.19) in a similar fashion. The monotonicity of the function f(x) = x<i>( -py"x) for x > 0 leads to As before, it is easily shown that (3.27) 1 x-1<!>( -py'x) dx = T 1 p 1 (x-112lnx) (py'x) dx The inequality = (2y"7r)-1{ roo (ln2y)y-112exp(-y)dy }p2j2 - roo 2(lnp)y-112 exp( -y) dy}. }p2j2 roo (ln2y)y-112 exp( -y) dy > roo (ln2y)y-112 exp( -y) dy }~~ h holds for p < 1. Therefore, (3.28) 1oo x-1<!>( -py'x) dx > C- 1f-1/2lnp 1oo y-1/2 exp( -y) dy p2/2 + 1f-1/2lnp 1 y-1/2 exp( -y) dy = C + ( -1 + h(p)) lnp
RANDOM WAKS AND EVES OF THE FORM n"' 133 where Cis a constant independent of p and h(p) = 1r-1/ 2 Jt' / 2 y-1/ 2 exp( -y) dy. We observe that (3.29) p2/2 h(p) < 7r-1/2fo y-1/2 dy = V(2/7r) p. Combining Corollary 3.1 and the results described in (3.20), (3.21), (3.26) and (3.28) yields that for p < 1, (3.30) E[T(c)IT(c) < oo] < (1/2)p-2( -1 + Cp-l+h(p))- 1 = (1/2)p-1 ( c ph(p) - p) -1 where C is a positive constant independent of p. By (3.29), the inequality ph(p) > p~p holds for p < 1. It follows from (3.30) that (3.31) limsuppe[t(c)it(c) < oo] < oo. p--+0 Combining the results of (3.25) and (3.31) proves the theorem. o Acknowledgments. This research was supported in part by NSF RED grant DMS 93555. REFERENCES 1. R.B. Ash, Real analysis and probability, Academic Press, New York, 1972. 2. W. Feller, An introduction to probability theory and its applications, Vol. I, 3rd ed., John Wiley and Sons, New York, 1968. 3. --,An introduction to probability theory and its applications, Vol. 2, 2nd ed., John Wiley and Sons, New York, 1971. 4. G.N. Griffiths and F.T. Wright, Moments of the number of deviations of sums of independent identically distributed random variables, Sankhya Ser. A. 37 (1975), 452-455. 5. T.. ai and K.K. an, On the last time and the number of boundary crossings related to the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 34 (1976), 59-71. 6. D. Razanadrakoto and N.C. Severo, The distribution of the 'finitely many times' in the strong law of large numbers, Statist. Probab. ett. 4 (1986), 179-182.
134 S. DRUCKER, I. FODOR AND E. FISHER 7. J. Slivka, On the law of the iterated logarithm, Proc. Nat. Acad. Sci. 63 (1969), 289-291. 8. ---, Some density functions of a counting variable in the simple random walk, Skand. Actuarietidskr. 53 (1970), 51-57. 9. J. Slivka and N.C. Severo, On the strong law of large numbers, Proc. Amer. Math. Soc. 24 (1970), 729-734. 10. W. Stout, Almost sure convergence, Academic Press, New York, 1974. 11. H.H. Stratton, Moments of oscillations and related sums, Ann. Math. Statist. 43 (1972), 1012-1016. UNIVERSITY OF CHICAGO, CHICAGO, I 60637 Current address: DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY, EAST ANSING, Ml 48824 RUTGERS UNIVERSITY, NEW BRUNSWICK, NJ 08903 Current address: DEPARTMENT OF STATISTICS, UNIVERSITY OF CAIFORNIA - BERKEEY, BERKEEY, CA 94720-7360 DEPARTMENT OF MATHEMATICS, AFAYETTE COEGE, EASTON, PA 18042-1781