On the weak law of large numbers for normed weighted sums of I.I.D. random variables
For weighted sums ∑j=1najYj of independent and identically distributed random variables {Yn,n≥1}, a general weak law of large numbers of the form (∑j=1najYj−νn)/bn→P0 is established where {νn,n≥1} and {bn,n≥1} are statable constants. The hypotheses involve both the behavior of the tail of the distri...
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| Format: | Article |
| Language: | English |
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Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171291000182 |
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| _version_ | 1832551309645971456 |
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| author | André Adler Andrew Rosalsky |
| author_facet | André Adler Andrew Rosalsky |
| author_sort | André Adler |
| collection | DOAJ |
| description | For weighted sums ∑j=1najYj of independent and identically distributed random variables
{Yn,n≥1}, a general weak law of large numbers of the form (∑j=1najYj−νn)/bn→P0 is established where
{νn,n≥1} and {bn,n≥1} are statable constants. The hypotheses involve both the behavior of the tail of the
distribution of |Y1| and the growth behaviors of the constants {an,n≥1} and {bn,n≥1}. Moreover, a weak
law is proved for weighted sums ∑j=1najYj indexed by random variables {Tn,n≥1}. An example is presented
wherein the weak law holds but the strong law fails thereby generalizing a classical example. |
| format | Article |
| id | doaj-art-2dc6518bbdcb491d85554ffef34e3483 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1991-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-2dc6518bbdcb491d85554ffef34e34832025-02-03T06:01:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251991-01-0114119120210.1155/S0161171291000182On the weak law of large numbers for normed weighted sums of I.I.D. random variablesAndré Adler0Andrew Rosalsky1Department of Mathematics, Illinois Institute of Technology, Chicago 60616, Illinois, USADepartment of Statistics, University of Florida, Gainesville, Florida, USAFor weighted sums ∑j=1najYj of independent and identically distributed random variables {Yn,n≥1}, a general weak law of large numbers of the form (∑j=1najYj−νn)/bn→P0 is established where {νn,n≥1} and {bn,n≥1} are statable constants. The hypotheses involve both the behavior of the tail of the distribution of |Y1| and the growth behaviors of the constants {an,n≥1} and {bn,n≥1}. Moreover, a weak law is proved for weighted sums ∑j=1najYj indexed by random variables {Tn,n≥1}. An example is presented wherein the weak law holds but the strong law fails thereby generalizing a classical example.http://dx.doi.org/10.1155/S0161171291000182weighted sums of independent and identically distributed random variables weak law of large numbersconvergence in probabilityrandom indicesstrong law of large numbersalmost certain convergence. |
| spellingShingle | André Adler Andrew Rosalsky On the weak law of large numbers for normed weighted sums of I.I.D. random variables International Journal of Mathematics and Mathematical Sciences weighted sums of independent and identically distributed random variables weak law of large numbers convergence in probability random indices strong law of large numbers almost certain convergence. |
| title | On the weak law of large numbers for normed weighted sums of I.I.D. random variables |
| title_full | On the weak law of large numbers for normed weighted sums of I.I.D. random variables |
| title_fullStr | On the weak law of large numbers for normed weighted sums of I.I.D. random variables |
| title_full_unstemmed | On the weak law of large numbers for normed weighted sums of I.I.D. random variables |
| title_short | On the weak law of large numbers for normed weighted sums of I.I.D. random variables |
| title_sort | on the weak law of large numbers for normed weighted sums of i i d random variables |
| topic | weighted sums of independent and identically distributed random variables weak law of large numbers convergence in probability random indices strong law of large numbers almost certain convergence. |
| url | http://dx.doi.org/10.1155/S0161171291000182 |
| work_keys_str_mv | AT andreadler ontheweaklawoflargenumbersfornormedweightedsumsofiidrandomvariables AT andrewrosalsky ontheweaklawoflargenumbersfornormedweightedsumsofiidrandomvariables |