A^VÇÚO 33 ò 3 Ï 207 c 6 Chinese Journal of Applied Probability Statistics Jun., 207, Vol. 33, No. 3, pp. 257-266 doi: 0.3969/j.issn.00-4268.207.03.004 Exact Asymptotics in Complete Moment Convergence for Record Times the Associated Counting Process KONG LingTao DAI HongShuai (School of Statistics, Shong University of Finance Economics, Jinan, 25004, China) Abstract: Let {X n, n } be a sequence of i.i.d. rom variables with absolutely continuous distribution function. Denote the record times the associated counting process of {X n, n } by {L(n), n } {µ(n), n }, respectively. In this paper, we obtain the exact asymptotics in complete moment convergence of {L(n), n } {µ(n), n }. Keywords: counting process; record times; exact asymptotics; complete moment convergence 200 Mathematics Subject Classification: 60F5; 60G50 Citation: Kong L T, Dai H S. Exact asymptotics in complete moment convergence for record times the associated counting process [J]. Chinese J. Appl. Probab. Statist., 207, 33(3): 257 266.. Introduction Let {X, X n ; n N} be a sequence of i.i.d. rom variables with continuous distribution function. And define the partial sum S n = given 0 < p < 2 r p, n k= X k, n N. It is well-known that, n r/p 2 P( S n ɛn /p ) <, ɛ > 0, () n= if only if E X r <, when r, E[X] = 0. For r = 2 p =, Hsu Robbins [] first proved the sufficiency. Later, Erdős [2, 3] obtained the necessity. For the special case r = p =, we refer to [4]. Baum Katz [5] obtained the general case. Dai was supported by the National Natural Science Foundation of China (Grant No. 36007), the Natural Science Foundation of Guangxi Province (Grant No. 204GXNSFCA800) the Fostering Project of Dominant Discipline Talent Team of Shong Province Higher Education Institutions. Kong was supported by the National Natural Science Foundation of China (Grant Nos. 76704; 60260; 62642), the Project of Humanities Social Science of Ministry of Education of China (Grant No. 6YJA90003) the Fostering Project of Dominant Discipline Talent Team of Shong University of Finance Economics. Corresponding author, E-mail: mathdsh@gmail.com. Received July 29, 205. Revised September 2, 205.
258 Chinese Journal of Applied Probability Statistics Vol. 33 Note that the sum in () tends to infinity as ɛ 0. Hence finding the precise rate at which this occurs becomes an interesting topic. In fact, it has been studied extensively. For example, Heyde [6] proved that ɛ2 n= P( S n ɛn) = E[X 2 ], if only if E[X] = 0 E[X 2 ] <. For more information on this topic, we refer to [7 2] so on. On the other h, based on (), another interesting topic is to study the complete moment convergence. Let p, α > /2, pα > E{ X p + X ln( + X )} <. Under the assumption that E[X] = 0, Chow [3] obtained that for any ɛ > 0, { n (p )α 2 E max S j ɛn α} <, j n + where {x} + = max{x, 0}. n= In this paper, the main aim is to study the exact asymptotics in complete moment convergence for the record times the associated counting process. introduce the corresponding definitions. Hence, we first Let {X, X n ; n N} be a sequence of i.i.d. rom variables with absolutely continuous distribution function. Set L() =, recursively, L(n) = min{k > L(n ) : X k > X L(n ) }, n 2. We call {L(n), n } as the record times of {X n, n }. The associated counting process {µ(n), n } is defined by µ(n) = max{k : L(k) n}. Many results about {µ(n)} have been established. For example, see [4 6] so on. Here, we should point out the recent results obtained by [7]. The results read as follows. Theorem ɛ ɛ 2 2r 2r (i) Let r > 0, then (ln ) r P { µ(n) > ɛ ln ln } = n (ii) When r = 0, we have ln ɛ n ln P{ µ(n) > ɛ ln } = 2 ɛ2 n ln P{ µ(n) > ɛ ln ln } =. 2 r.
No. 3 KONG L. T., DAI H. S.: Exact Asymptotics in Complete Moment Convergence 259 Motivated by the above results, we aim to investigate the exact asymptotics in complete moment convergence for {L(n), n N} {µ(n), n N} in this work. The rest of this paper is organized as follows. In Section 2, we list some basic facts about the record times the associated counting process. Furthermore, we obtain the exact asymptotics in complete moment convergence for the associated counting process. Section 3 is devoted to getting the exact asymptotics for the record times. Throughout this paper, we use C to denote an positive constant whose value may vary from line to the next. The notation f(x) g(x) means that f(x)/g(x) as x. 2. Exact Asymptotics for the Counting Process In this section, we study the exact asymptotics for the counting process. In order to reach our aim, we first recall some basic facts about the record times the counting process. Let, if X k is a record, I k = 0, otherwise, then µ(n) = n I k, n. It follows from [5] that, as n, (i) k= m n = Eµ(n) = n /k = + γ + o(), k= (ii) Var µ(n) = n ( /k)/k = + γ π 2 /6 + o(), k= (iii) µ(n)/ a.s., (iv) [µ(n) ]/ (v) ln L(n)/n a.s., (vi) (ln L(n) n)/ n d N(0, ), d N(0, ), where γ = 0.577, is the Euler s constant. Now, we state the main results of this section. We have Theorem 2 Given r > 0, (ln ) r n() 3/2 E{ µ(n) ɛ ln ln } + = r 2π.
260 Chinese Journal of Applied Probability Statistics Vol. 33 Theorem 3 When r = 0, ln ɛ ɛ2 Remark 4 n() 3/2 ln E{ µ(n) ɛ ln } = 2E N, (2) + n() 3/2 ln E{ µ(n) ɛ ln ln } + = 3 E N 3. (3) Zang Fu [8] established the exact asymptotics in complete moment convergence for the counting process. However, they dealt with the case of large deviation, i.e.,, in this paper, we investigate the case of moderate deviation, i.e., ln, in (2), the case of small deviation, i.e., ln ln, in Theorem 2 the equaiton (3). Remark 5 When we checked the proof of this article, we found by chance that some similar results were obtained by [9] [20]. They used the truncated method to estimate the remainder terms. However, in the proofs of our results, we mainly rely on the estimate of n, which is given by Lemma 9. In order to prove Theorem 2, we need some technical results. We have the following propositions. Proposition 6 Given r > 0, we have (ln ) r n E { N ɛ ln ln } + = r 2π. Before we prove it, we need a technical lemma, which comes from [2]. Lemma 7 For large enough x, Ψ(x) = 2P(N x) 2 2πx e x2 /2, where Ψ(x) = P( N x) with N being the stard normal rom variable. Next, we prove the Proposition 6. Proof of Proposition 6 Note that Γ = Γ = For convenience, let (ln ) r E { N ɛ ln ln } n +. (ln ln x) r 9 x ln x ɛ Ψ(y)dydx. (4) ln ln ln x
No. 3 KONG L. T., DAI H. S.: Exact Asymptotics in Complete Moment Convergence 26 Let t = ɛ ln ln ln x. Then (4) is equivalent to Γ = = = = (5) Lemma 7 imply that Γ = r ln(ɛ 2 2r) r ln(ɛ 2 2r) 2t ɛ 2 ert2 /ɛ 2 Ψ(y) r ln(ɛ 2 2r) Let s = (ɛ 2 2r)y 2 /(ɛ 2 ln ln ln 9). Then Γ = r 2π ln(ɛ 2 2r) ɛ 2 2r y t Ψ(y)dydt 2t ɛ 2 ert2 /ɛ 2 dtdy Ψ(y)(e ry2 /ɛ 2 e r ln ln ln 9 )dy Ψ(y)e ry2 /ɛ 2 dy. (5) 2 2πy e (ɛ2 2r)y 2 /(2ɛ 2) dy. s e s ln ln ln 9/2 ds = r 2π. We complete the proof of Proposition 6. Proposition 8 Let A(n) = E { µ(n) ɛ ln ln } + E { N ɛ ln ln }. + Then, for any r > 0, (ln ) r A(n) = 0. n() 3/2 In order to prove it, we also need a technical lemma. introduce the following notation. For any x 0, let n = sup x Following [22], we have ( P( µ n > x) P N > Before we state it, we first x ). (6) Lemma 9 n 3.8, n 2, where n is defined by (6).
262 Chinese Journal of Applied Probability Statistics Vol. 33 Next, we prove the Proposition 8. Proof of Proposition 8 that, for any r > 0, For convenience, let Note that Λ = In order to prove the proposition, we only need to prove (ln ) r A(n) <. n() 3/2 Λ = (ln ) r n() 3/2 ɛ ln ln (ln ) r A(n). n() 3/2 ɛ ln ln P( N t)dt. By using the change of variable u = x/, we have where I = I 2 = Λ = =: / n 0 / n (ln ) r n P ( N u + ɛ ln ln )] du (ln ) r n 0 P ( N u + ɛ ln ln ) du By Lemma 9, we have Note that (ln ) r (I + I 2 ), n 0 P( µ(n) x)dx [ ( µ(n) ) P u + ɛ ln ln ( µ(n) ) P u + ɛ ln ln ( µ(n) ) P u + ɛ ln ln P ( N u + ɛ ln ln ) du ( µ(n) ) P u + ɛ ln ln P ( N u + ɛ ln ln ) du. I n µ(n) C. (7) () /4 d N(0, ). Thus, Lemma 7 implies that, as u +, ( µ(n) ) P u C /2 u e u2 (8)
No. 3 KONG L. T., DAI H. S.: Exact Asymptotics in Complete Moment Convergence 263 P( N u) C u e u2 /2. (9) On the other h, we have u e u2 du, y as y +. (0) Combining (8), (9) (0), we get I 2 y Thus, by (7) (), we have / n C u e u2 /2 du C n Λ C. () () /4 (ln ) r <. (2) n() 5/4 We complete the proof of Proposition 8. Remark 0 By (2), Λ is also finite when r 0. Now we st at a point where we can prove the theorem 2. Proof of Theorem 2 triangle inequality directly. Theorem 2 follows from Propositions 6 8, the Next, we prove Theorem 3. Similar to the proof of Theorem 2, in order to prove it, we also need the following technical propositions. Proposition Proof ln ɛ ɛ2 some calculations, we have n ln E{ N ɛ ln } = 2E N, (3) + n ln E{ N ɛ ln ln } + = 3 E N 3. (4) The proof of Proposition is similar to that of Proposition 6. In fact, by ln ɛ = ln ɛ 2 = ln ɛ = 2E N, 3 n ln E{ N ɛ ln } + Ψ(y)dydx x ln x ln ln x ɛ ln ln 3 ɛ ln ln x (ln y ln ɛ ln ln ln 3 )Ψ(y)dy
264 Chinese Journal of Applied Probability Statistics Vol. 33 ɛ2 ɛ 2 9 = = where we used the fact that n ln E{ N ɛ ln ln } + Ψ(y)dydx x ln x ln ln x We complete the proof of Proposition. Then Proposition 2 Let ɛ ln ln ln x y 2 Ψ(y)dy = 3 E N 3, 2 ln ɛ ɛ ln yψ(y)dy = 0. ln ln 3 B(n) = E { µ(n) ɛ ln } + E { N ɛ ln }. + ln ɛ n() 3/2 B(n) = 0. ln Proposition 2 can be proved in the same way as the proof of Proposition 8. We omit the details here. Next, we prove Theorem 3. Proof of Theorem 3 It follows from (3), Proposition 2 the triangle inequality that (2) holds. On the other h, by (4), Remark 0 the triangle inequality, we obtain (3). The proof of Theorem 3 is finished. 3. Exact Asymptotics for the Record Times In this section, we study the precise asymptotics for the record times {L(n) : n N}. The following results for record times can be proved in the same way as the proof of Theorem 2 with the estimate given by Theorem 4 in [22]. Theorem 3 ɛp(δ+) Theorem 4 ɛ2(δ+) Given δ > p > 0, we have n δ E { ln L(n) n ɛn /p+/2} + = Given δ >, we have E N pδ+p+ p(δ + )(pδ + p + ). () δ n 3/2 E{ ln L(n) n ɛ n } + = E N 2δ+3 (δ + )(2δ + 3).
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