Compound Poisson approximation to weighted sums of symmetric discrete variables

The weighted sum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$S=w_1S_1+w_2S_2+\cdots +w_NS_N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mo>=</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>w</mi> <mi>N</mi> </msub> <msub> <mi>S</mi> <mi>N</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation> is approximated by compound Poisson distribution. Here <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$S_i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>S</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> are sums of symmetric independent identically distributed discrete random variables, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$w_i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>w</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> denote weights. The estimates take into account the smoothing effect that sums <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$S_i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>S</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> have on each other. Copyright The Institute of Statistical Mathematics, Tokyo 2015