TY - JOUR

T1 - Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables

AU - Grinblat, L. S.

PY - 1977

Y1 - 1977

N2 - Let €(t),…» €n(t)»… and (t) be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of ƒ(€(t)) converges to the distribution of ƒ(€ (t)) as ƒ i->oo for any functional ƒ continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions € (t) T (0 < t < 1) without discontinuities of the second kind. For a set T» {t1, C [0, 1] the metric pT is defined on as in 2.3. The metric pT defines on the X the minimal topology in which all functionals continuous in Skorohod’s metric and also the functionals x(t1 - 0), x(t1),…»x(tn - 0), *(tn),… are continuous. We will give necessary and sufficient conditions under which the distribution of ƒ(€n(t)) converges to the distribution of ƒ (€ (t)) as n-> oo for any completely continuous functional ƒ, i.e. for any functional ƒ which is continuous in any of the metrics pT defined in 2.3.

AB - Let €(t),…» €n(t)»… and (t) be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of ƒ(€(t)) converges to the distribution of ƒ(€ (t)) as ƒ i->oo for any functional ƒ continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions € (t) T (0 < t < 1) without discontinuities of the second kind. For a set T» {t1, C [0, 1] the metric pT is defined on as in 2.3. The metric pT defines on the X the minimal topology in which all functionals continuous in Skorohod’s metric and also the functionals x(t1 - 0), x(t1),…»x(tn - 0), *(tn),… are continuous. We will give necessary and sufficient conditions under which the distribution of ƒ(€n(t)) converges to the distribution of ƒ (€ (t)) as n-> oo for any completely continuous functional ƒ, i.e. for any functional ƒ which is continuous in any of the metrics pT defined in 2.3.

UR - http://www.scopus.com/inward/record.url?scp=84966210841&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1977-0494376-9

DO - 10.1090/S0002-9947-1977-0494376-9

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AN - SCOPUS:84966210841

SN - 0002-9947

VL - 234

SP - 361

EP - 379

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 2

ER -