Variance of partial sums of stationary sequences
Let $X_1,X_2,\ldots$ be a centred sequence of weakly stationary random variables with spectral measure $F$ and partial sums $S_n=X_1+\cdots+X_n$. We show that $\operatorname {var}(S_n)$ is regularly varying of index $\gamma$ at infinity, if and only if $G(x):=\int_{-x}^xF(\mathrm {d}x)$ is regularly...
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| Format: | Journal article |
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Institute of Mathematical Statistics
2012
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| _version_ | 1826259091043909632 |
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| author | Deligiannidis, G Utev, S |
| author_facet | Deligiannidis, G Utev, S |
| author_sort | Deligiannidis, G |
| collection | OXFORD |
| description | Let $X_1,X_2,\ldots$ be a centred sequence of weakly stationary random variables with spectral measure $F$ and partial sums $S_n=X_1+\cdots+X_n$. We show that $\operatorname {var}(S_n)$ is regularly varying of index $\gamma$ at infinity, if and only if $G(x):=\int_{-x}^xF(\mathrm {d}x)$ is regularly varying of index $2-\gamma$ at the origin ($0<\gamma<2$). |
| first_indexed | 2024-03-06T18:44:23Z |
| format | Journal article |
| id | oxford-uuid:0e013088-cfd0-4dea-9141-d0ac8f1be78a |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:44:23Z |
| publishDate | 2012 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:0e013088-cfd0-4dea-9141-d0ac8f1be78a2022-03-26T09:43:31ZVariance of partial sums of stationary sequencesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:0e013088-cfd0-4dea-9141-d0ac8f1be78aSymplectic Elements at OxfordInstitute of Mathematical Statistics2012Deligiannidis, GUtev, SLet $X_1,X_2,\ldots$ be a centred sequence of weakly stationary random variables with spectral measure $F$ and partial sums $S_n=X_1+\cdots+X_n$. We show that $\operatorname {var}(S_n)$ is regularly varying of index $\gamma$ at infinity, if and only if $G(x):=\int_{-x}^xF(\mathrm {d}x)$ is regularly varying of index $2-\gamma$ at the origin ($0<\gamma<2$). |
| spellingShingle | Deligiannidis, G Utev, S Variance of partial sums of stationary sequences |
| title | Variance of partial sums of stationary sequences |
| title_full | Variance of partial sums of stationary sequences |
| title_fullStr | Variance of partial sums of stationary sequences |
| title_full_unstemmed | Variance of partial sums of stationary sequences |
| title_short | Variance of partial sums of stationary sequences |
| title_sort | variance of partial sums of stationary sequences |
| work_keys_str_mv | AT deligiannidisg varianceofpartialsumsofstationarysequences AT utevs varianceofpartialsumsofstationarysequences |