The integral limit theorem in the first passage problem for sums of independent nonnegative lattice variables
<p>The integral limit theorem as to the probability distribution of the random number <mml:math> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> of summands in the sum <mml:math> <mml:mstyl...
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| Format: | Article |
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| Language: | English |
| Published: |
Wiley
2006-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/56367 |
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| Summary: | <p>The integral limit theorem as to the probability distribution of the random number <mml:math> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> of summands in the sum <mml:math> <mml:mstyle displaystyle='true'> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:mstyle> </mml:math> is proved. Here, <mml:math> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo><mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo><mml:mo>…</mml:mo> </mml:math> are some nonnegative, mutually independent, lattice random variables being equally distributed and <mml:math> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> is defined by the condition that the sum value exceeds at the first time the given level <mml:math> <mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℕ</mml:mi> </mml:math> when the number of terms is equal to <mml:math> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math>.</p> |
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| ISSN: | 1085-3375 |