Limit theorems for variational sums
Authors:
William N. Hudson and Howard G. Tucker
Journal:
Trans. Amer. Math. Soc. 191 (1974), 405426
MSC:
Primary 60F05
MathSciNet review:
0358928
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Limit theorems in the sense of a.s. convergence, convergence in norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form is shown to converge with probability one and in norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.
 [1]
Simeon
M. Berman, Signinvariant random variables and
stochastic processes with signinvariant increments, Trans. Amer. Math. Soc. 119 (1965), 216–243. MR 0185651
(32 #3113), http://dx.doi.org/10.1090/S00029947196501856518
 [2]
R.
M. Blumenthal and R.
K. Getoor, Sample functions of stochastic processes with stationary
independent increments, J. Math. Mech. 10 (1961),
493–516. MR 0123362
(23 #A689)
 [3]
Robert
Cogburn and Howard
G. Tucker, A limit theorem for a function of the
increments of a decomposable process, Trans.
Amer. Math. Soc. 99
(1961), 278–284. MR 0123353
(23 #A681), http://dx.doi.org/10.1090/S00029947196101233530
 [4]
Priscilla
Greenwood and Bert
Fristedt, Variations of processes with stationary, independent
increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
23 (1972), 171–186. MR 0321197
(47 #9730)
 [5]
Michel
Loève, A l’intérieur du problème
central, Publ. Inst. Statist. Univ. Paris 6 (1957),
313–325 (French). MR 0100911
(20 #7336)
 [6]
Michel
Loève, Probability theory, Third edition, D. Van
Nostrand Co., Inc., Princeton, N.J.Toronto, Ont.London, 1963. MR 0203748
(34 #3596)
 [7]
P.
W. Millar, Path behavior of processes with stationary independent
increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
17 (1971), 53–73. MR 0324781
(48 #3130)
 [8]
E.
L. Rvačeva, On domains of attraction of multidimensional
distributions, L ′vov. Gos. Univ. Uč. Zap. Ser. Meh.Mat.
29 (1954), no. 6, 5–44 (Russian). MR 0076211
(17,864e)
 [9]
Howard
G. Tucker, On asymptotic independence of the partial sums of
positive and negative parts of independent random variables, Advances
in Appl. Probability 3 (1971), 404–425. MR 0288822
(44 #6017)
 [10]
Howard
G. Tucker, A graduate course in probability, Probability and
Mathematical Statistics, Vol. 2, Academic Press, Inc., New YorkLondon,
1967. MR
0221541 (36 #4593)
 [11]
Stephen
James Wolfe, On moments of infinitely divisible distribution
functions, Ann. Math. Statist. 42 (1971),
2036–2043. MR 0300319
(45 #9365)
 [1]
 S. M. Berman, Signinvariant random variables and stochastic processes with signinvariant increments, Trans. Amer. Math. Soc. 119 (1965), 216243. MR 32 #3113. MR 0185651 (32:3113)
 [2]
 R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), 493516. MR 23 #A689. MR 0123362 (23:A689)
 [3]
 R. Cogburn and H. G. Tucker, A limit theorem for a function of the increments of a decomposable process, Trans. Amer. Math. Soc. 99 (1961), 278284. MR 23 #A681. MR 0123353 (23:A681)
 [4]
 P. Greenwood and B. Fristedt, Variations of processes with stationary independent increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 171186. MR 0321197 (47:9730)
 [5]
 M. Loève, A l'intérieur du problème central, Publ. Inst. Statist. Univ. Paris (homage à M. Paul Lévy) 6 (1957), 313325. MR 20 #7336. MR 0100911 (20:7336)
 [6]
 , Probability theory. Foundations. Random sequences, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1963. MR 34 #3596. MR 0203748 (34:3596)
 [7]
 P. W. Millar, Path behavior of processes with stationary independent increments, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 5373. MR 0324781 (48:3130)
 [8]
 E. L. Rvačeva, On domains of attraction of multidimensional distributions, L'vov. Gos. Univ. Uč. Zap. 29, Ser. Meh.Mat. No. 6 (1954), 544; English transl., Selected Transl. Math Statist. and Probability, vol. 2, Amer. Math. Soc., Providence, R. I., 1962, pp. 183205. MR 17, 864; 27 #782. MR 0076211 (17:864e)
 [9]
 H. G. Tucker, On asymptotic independence of the partial sums of the positive parts and negative parts of independent random variables, Advances in Appl. Probability 3 (1971), 404425. MR 44 #6017. MR 0288822 (44:6017)
 [10]
 , A graduate course in probability, Probability and Math. Statist., vol. 2, Academic Press, New York and London, 1967. MR 36 #4593. MR 0221541 (36:4593)
 [11]
 S. J. Wolfe, On moments of infinitely divisible distributions, Ann. Math. Statist. 42 (1971), 20362043. MR 0300319 (45:9365)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
60F05
Retrieve articles in all journals
with MSC:
60F05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403589286
PII:
S 00029947(1974)03589286
Keywords:
Variational sums,
infinitely divisible distributions,
stochastic processes with independent increments
Article copyright:
© Copyright 1974
American Mathematical Society
