A note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights

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1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at A note on almost sure behavor of randomly weghted sums of φ-mxng random varables wth φ-mxng weghts Marcn Przystalsk Abstract. Randomly weghted sums play an mportant role n varous appled and theoretcal problems, e.g., n actuaral mathematcs or statstcs. The almost sure convergence of randomly weghted sums s usually studed under the assumpton that sequences are ndependent and dentcally dstrbuted. In ths note, we assume that both sequences are φ-mxng. Under some addtonal condtons, we prove a strong law of large numbers for sequences of randomly weghted sums.. Introducton Let {Y, } be a sequence of random varables defned on a probablty space (Ω, F, P ). Let Fj k be a σ-algebra generated by the random varables Y l, l j,..., k. Defne the φ-mxng coeffcent (unform mxng coeffcent) φ (m) sup { P (B A) P (B) }, where the supremum s taken over A F k, B F k+m n, P (A) 0, k n m. The unform mxng coeffcent was ntroduced ndependently by Rozanov and Volkonsk [] and Ibragmov [4]. Snce then many authors have studed sequences of φ-mxng random varables, and a number of useful results have been establshed. In [8], Nagaev proved probablty and maxmal nequaltes for φ-mxng random varables. The summablty of φ-mxng random varables was studed by Kesel n [5, 6], whereas strong laws of large numbers were obtaned, e.g., n [5, 6, 7, 2]. Receved May 2, Mathematcs Subject Classfcaton. 60F5. Key words and phrases. Randomly weghted sums, φ-mxng sequences of random varables, strong law of large numbers. 27

2 28 MARCIN PRZYSTALSKI Randomly weghted partal sums n A X play an mportant role n varous appled and theoretcal problems. In actuaral mathematcs, f X s regarded as the net loss wthn the tme perod of the company and each A as the dscount factor from tme to tme 0, then n A X can be nterpreted as the total dscounted amount of the net loss from tme 0 to tme n. For example, n the feld of queueng theory, the n A X can be used to represent the total output for customer beng served by n machnes. In statstcs, Arenal-Gutérez et al. [] obtaned a strong law of large numbers for the bootstrap mean, assumng that {X, } s a sequence of parwse ndependent and dentcally dstrbuted random varables. The latter was further generalzed by Rosalsky and Sreehar n [0]. In ths note, we study strong lmt theorems of randomly weghted partal sums n A X, assumng that both sequences are weakly dependent, whch generalze the results obtaned n [, 0]. In contrast to [], we assume that {X, } s φ-mxng. On the sequence of weghts {A, }, we assume that ths sequence s a sequence of postve, dentcally dstrbuted (.d.) φ-mxng random varables such that A and X are ndependent, for each. We establsh strong laws of large numbers for a non-dentcally dstrbuted sequence {X, } usng the noton of a regular cover, whch was ntroduced by Pruss n [9]. Defnton.. Let X,..., X n be random varables, and X be a random varable possbly defned on a dfferent probablty space. Then X,..., X n are sad to be a regular cover of X provded E (G (X)) E (G (X )), () n for any measurable functon G for whch both sdes make sense. 2. Techncal lemmas Let {Y, } be a sequence of φ-mxng random varables. Set S k k Y and M n max k n S k. Defne φ + (m) sup {P (B A) P (B)}, where the supremum s taken over A F k, B F k+m n, P (A) 0, k n m. In [8], t was ponted out that φ + (n) < φ (n). Assume that φ + () < and let δ > 0 satsfy the condton δ + φ + () <. Set ρ δ + φ + (). Let α be a number such that the followng condton s satsfed: P (2M n > α) < δ. Under the above notaton, Nagaev [8] proved the followng nequalty.

3 RANDOMLY WEIGHTED SUMS 29 Lemma 2.. For any p > 0 and 0 < ε < 6 such that s (ε) > p, EMn p < c (p) E Y p + c 2 (p) α p, where s (ε) log ρ/ log ( + ε), c (p) < 2p+ ε 3p+ B (p +, s (ε) p + ), ρ c 2 (p) < ρ ε p B (p +, s (ε) p) p +, and B (, ) s the Euler Beta functon. In the proof of the man result we wll need the followng lemma. Lemma 2.2. Let {A, A, } be a sequence of postve.d. varables wth EA p <, for some p 2. Then max n A n 0 a.s. random Proof. Note that the condton EA p < mples n P (A n > ɛn) <, for every ɛ > 0. Hence, by the Borel Cantell lemma we have that A n n 0 a.s. Thus, by Lemma n [3] we get the asserton. Throughout ths paper, C and C 2 always stand for postve constants whch may dffer from one place to another. 3. Man results Theorem 3.. Let {A, A, } be a sequence of postve.d. φ-mxng random varables wth EA p <, for some p 2, and let {X, } be a sequence of φ-mxng random varables that s ndependent of {A, A, }. Let X be a random varable, possbly defned on a dfferent probablty space, satsfyng condton (). Moreover, addtonally assume that EX n 0, for all n. Let b 0 0 and be an ncreasng sequence of postve numbers satsfyng n and bp n O (n). (2) b p n If s (ε) > p, for some 0 < ε < 6, and E X p <, then lm n n /p A X 0 a.s. (3)

4 30 MARCIN PRZYSTALSKI Proof. For n, set X X I ( X b ) and X X I ( X > b ). Then A X ( A X EX ) + A X + A EX. In order to show that ( n /p ) n A X 0 a.s., we only need to show that all terms above are o ( n /p ) a.s. Frst, we show that ( ( P A X EX ) ) > n /p ɛ <, (4) n for all ɛ > 0. Because X X I ( X b ) s also φ-mxng, by Theorem 5.2 n [2], we have that {A X, } s also φ-mxng. Hence, by Markov s nequalty and Lemma 2., we have that ( ( P A X EX ) ) >n /p ɛ n n C E n A (X EX ) p ɛ p n + C 2 C EA p n n I + I 2. E X EX p E X p + C 2 n Note that the second part of (2) ensures that I 2 <. Hence, t remans to show that I <. Because s ncreasng, we have that b, for all n, and I C C n n E X p E X p I ( X ).

5 RANDOMLY WEIGHTED SUMS 3 Let G (x) x p I [ x ]; then by the defnton of regular cover I C C where X XI ( X ). Further, I C C n n n E X p I ( X ) E X p, E X p C n n n b p P (b < X b ) E X p I [b < X b ] C b p P (b < X b ). (5) Then, by (5) and (2), n I C P (b < X b ) C P ( X > b ) C E X p <, b p and (4) holds. Thus, ( n /p ) n A (X EX ) converges completely to 0, whch mples that n A (X EX ) s o ( n /p ) a.s. Next, by the defnton of regular cover, condton E X p <, and (2), we have that P ( X n > ) P (b < X n b ) n n n n n P (b < X n b ) EI [b < X n b ] EI [b < X b ]

6 32 MARCIN PRZYSTALSKI P (b < X b ) E X p b p <. Hence, by the Borel Cantell lemma, n X s bounded a.s. By (2) and Lemma 2.2, t follows that n /p A X n /p max b A X n n n /p max n A X 0 a.s. n Fnally, by Markov s nequalty, Lemma 2., condton EA p <, and the defnton of regular cover, ( ) P A EX > n/p ɛ C EA p EX p + C2 n C n n E X p + C2 n n Hence, usng the same arguments as n the estmaton of I and I 2, we obtan that ( ) P A EX > n/p ɛ <, n for every ɛ > 0. Thus, ( n /p ) n A EX converges completely to 0, whch mples that n A EX s o ( n /p ) a.s. Ths completes the proof. In [9], t was ponted out that every.d. sequence of random varables satsfes the regular cover condton () wth X X. Thus, from Theorem 3. we have the followng corollary. Corollary 3.2. Let {A, A, } be a sequence of postve.d. φ-mxng random varables wth EA p <, for some p 2, and let {X, X, } be a sequence of.d. φ-mxng random varables that s ndependent of {A, A, }. Moreover, addtonally assume that EX n 0, for all n. Let b 0 0 and be an ncreasng sequence of postve numbers satsfyng (2). If s (ε) > p, for some 0 < ε < 6, and E X p <, then (3) holds. It s known that every sequence of ndependent and dentcally dstrbuted (..d.) random varables s φ-mxng wth φ (n) 0, for each n. Thus, from Theorem 3. we obtan the followng corollary..

7 RANDOMLY WEIGHTED SUMS 33 Corollary 3.3. Let {A, A, } be a sequence of postve..d. random varables wth EA p <, for some p 2, and let {X, } be a sequence of φ-mxng random varables that s ndependent of {A, A, }. Let X be a random varable, possbly defned on a dfferent probablty space, satsfyng condton (). Moreover, addtonally assume that EX n 0, for all n. Let b 0 0 and be an ncreasng sequence of postve numbers satsfyng (2). If s (ε) > p, for some 0 < ε < 6, and E X p <, then (3) holds. We conclude wth some remarks. Remark. Usng Theorem 5.2 n [2], under some addtonal condtons, one can obtan the counterpart of Theorem 3. for other mxng coeffcents. Remark 2. It should be stressed that the assumpton of ndependence of {A, } and {X, } n Theorem 3. s very crucal. Assumng only that both sequences are φ-mxng does not guarantee that {A X, } wll be φ-mxng (see [2, Theorem 5.2], and dscusson below the theorem). References [] E. Arenal-Gutérez, C. Matrán, and J. A. Cuesta-Albertos, On the uncondtonal strong law of large numbers for bootstrap mean, Statst. Probab. Lett. 27 (996), [2] R. C. Bradley, Basc propertes of strong mxng condtons. A survey and some open questons, Probab. Surv. 2 (2005), [3] J. H. J. Enmahl and A. Rosalsky, General weak laws of large numbers for bootstrap sample means, Stochastc Anal. Appl. 23 (2005), [4] I. A. Ibragmov, Some lmt theorems for statonary processes, Theory Probab. Appl. 7 (962), [5] R. Kesel, Strong laws and summablty for sequences of φ-mxng random varables n Banach spaces, Electron. Comm. Probab. 2 (997), [6] R. Kesel, Summablty and strong laws of φ-mxng sequences, J. Theoret. Probab. (998), [7] A. Kuczmaszewska, On the strong law of large numbers for φ-mxng and ρ-mxng random varables, Acta Math. Hungar. 32 (20), [8] S. V. Nagaev, On probablty and moment nequaltes for dependent random varables, Theory Probab. Appl. 45 (2000), [9] A. R. Pruss, Randomly sampled Remann sums and complete convergence n the law of large numbers for a case wthout dentcal dstrbutons, Proc. Amer. Math. Soc. 24 (996), [0] A. Rosalsky and M. Sreehar, On the lmtng behavor of randomly weghted partal sums, Statst. Probab. Lett. 40 (998), [] Y. A. Rozanov and V. A. Volkonsk, Some lmt theorems for random functon, Theory Probab. Appl. 4 (959), [2] D. Q. Tuyen, A strong law of φ-mxng random varables, Perod. Math. Hungar. 38 (999), Research Center for Cultvar Testng, S lupa Welka, Poland E-mal address: marprzyst@gmal.com