Local limit theorem for the maximum of a random walk
Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]
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| creator | Kugler, Johannes |
| description | Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le
a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose
that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2] |
| doi_str_mv | 10.48550/arxiv.1403.7372 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_1403_7372</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1403_7372</sourcerecordid><originalsourceid>FETCH-LOGICAL-a652-bdfb24bb8b230965cf836294498ed57e71bad427d44783199d5c51c85943f6723</originalsourceid><addsrcrecordid>eNotzrFuwjAUhWEvDBV074T8Aklj-zq2R4SgVIrUhT26jm1hEWNkoNC3r0I7nX86-gh5Y00NWsrmHcsjftcMGlErofgLgS4PONIxpnil14PPxScacpmaJnzEdEs0B4q04MnlRO84HhdkFnC8-Nf_nZP9drNf76ru6-NzveoqbCWvrAuWg7XactGYVg5Bi5YbAKO9k8orZtEBVw5AacGMcXKQbNDSgAit4mJOln-3T3V_LjFh-eknfT_pxS-JHT26</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Local limit theorem for the maximum of a random walk</title><source>arXiv.org</source><creator>Kugler, Johannes</creator><creatorcontrib>Kugler, Johannes</creatorcontrib><description>Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le
a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose
that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]<\infty$ and $\sup_{a\le
a_0}\mathbf{E}[\max\{0,X^{(a)}\}^{2+\varepsilon}]<\infty$ for some
$\varepsilon>0$. Assume that $X^{(a)}\xrightarrow[]{w} X^{(0)}$ as $a\to 0$ and
denote by $M^{(a)}=\max_{k\ge 0} S_k^{(a)}$ the maximum of the random walk
$S^{(a)}$. In this paper we provide the asymptotics of
$\mathbf{P}(M^{(a)}=y\Delta)$ as $a\to 0$ in the case, when $y\to \infty$ and
$ay=O(1)$. This asymptotics follows from a representation of
$\mathbf{P}(M^{(a)}=y\Delta)$ via a geometric sum and a uniform renewal
theorem, which is also proved in this paper.</description><identifier>DOI: 10.48550/arxiv.1403.7372</identifier><language>eng</language><subject>Mathematics - Probability</subject><creationdate>2014-03</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1403.7372$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1403.7372$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Kugler, Johannes</creatorcontrib><title>Local limit theorem for the maximum of a random walk</title><description>Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le
a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose
that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]<\infty$ and $\sup_{a\le
a_0}\mathbf{E}[\max\{0,X^{(a)}\}^{2+\varepsilon}]<\infty$ for some
$\varepsilon>0$. Assume that $X^{(a)}\xrightarrow[]{w} X^{(0)}$ as $a\to 0$ and
denote by $M^{(a)}=\max_{k\ge 0} S_k^{(a)}$ the maximum of the random walk
$S^{(a)}$. In this paper we provide the asymptotics of
$\mathbf{P}(M^{(a)}=y\Delta)$ as $a\to 0$ in the case, when $y\to \infty$ and
$ay=O(1)$. This asymptotics follows from a representation of
$\mathbf{P}(M^{(a)}=y\Delta)$ via a geometric sum and a uniform renewal
theorem, which is also proved in this paper.</description><subject>Mathematics - Probability</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzrFuwjAUhWEvDBV074T8Aklj-zq2R4SgVIrUhT26jm1hEWNkoNC3r0I7nX86-gh5Y00NWsrmHcsjftcMGlErofgLgS4PONIxpnil14PPxScacpmaJnzEdEs0B4q04MnlRO84HhdkFnC8-Nf_nZP9drNf76ru6-NzveoqbCWvrAuWg7XactGYVg5Bi5YbAKO9k8orZtEBVw5AacGMcXKQbNDSgAit4mJOln-3T3V_LjFh-eknfT_pxS-JHT26</recordid><startdate>20140328</startdate><enddate>20140328</enddate><creator>Kugler, Johannes</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20140328</creationdate><title>Local limit theorem for the maximum of a random walk</title><author>Kugler, Johannes</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a652-bdfb24bb8b230965cf836294498ed57e71bad427d44783199d5c51c85943f6723</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Mathematics - Probability</topic><toplevel>online_resources</toplevel><creatorcontrib>Kugler, Johannes</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Kugler, Johannes</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Local limit theorem for the maximum of a random walk</atitle><date>2014-03-28</date><risdate>2014</risdate><abstract>Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le
a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose
that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]<\infty$ and $\sup_{a\le
a_0}\mathbf{E}[\max\{0,X^{(a)}\}^{2+\varepsilon}]<\infty$ for some
$\varepsilon>0$. Assume that $X^{(a)}\xrightarrow[]{w} X^{(0)}$ as $a\to 0$ and
denote by $M^{(a)}=\max_{k\ge 0} S_k^{(a)}$ the maximum of the random walk
$S^{(a)}$. In this paper we provide the asymptotics of
$\mathbf{P}(M^{(a)}=y\Delta)$ as $a\to 0$ in the case, when $y\to \infty$ and
$ay=O(1)$. This asymptotics follows from a representation of
$\mathbf{P}(M^{(a)}=y\Delta)$ via a geometric sum and a uniform renewal
theorem, which is also proved in this paper.</abstract><doi>10.48550/arxiv.1403.7372</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Probability |
| title | Local limit theorem for the maximum of a random walk |
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