The fixed points of the multivariate smoothing transform
Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\li...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Mentemeier, Sebastian [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag Berlin Heidelberg 2015 |
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| Übergeordnetes Werk: |
Enthalten in: Probability theory and related fields - Springer Berlin Heidelberg, 1986, 164(2015), 1-2 vom: 21. Jan., Seite 401-458 |
|---|---|
| Übergeordnetes Werk: |
volume:164 ; year:2015 ; number:1-2 ; day:21 ; month:01 ; pages:401-458 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00440-015-0615-y |
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OLC2054639710 |
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| 520 | |a Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. | ||
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10.1007/s00440-015-0615-y doi (DE-627)OLC2054639710 (DE-He213)s00440-015-0615-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Mentemeier, Sebastian verfasserin aut The fixed points of the multivariate smoothing transform 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. Smoothing transform Markov random walks General branching processes Multivariate stable laws Multitype branching random walk Choquet–Deny lemma Weighted branching Enthalten in Probability theory and related fields Springer Berlin Heidelberg, 1986 164(2015), 1-2 vom: 21. Jan., Seite 401-458 (DE-627)129382779 (DE-600)165783-5 (DE-576)01476914X 0178-8051 nnns volume:164 year:2015 number:1-2 day:21 month:01 pages:401-458 https://doi.org/10.1007/s00440-015-0615-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 164 2015 1-2 21 01 401-458 |
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10.1007/s00440-015-0615-y doi (DE-627)OLC2054639710 (DE-He213)s00440-015-0615-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Mentemeier, Sebastian verfasserin aut The fixed points of the multivariate smoothing transform 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. Smoothing transform Markov random walks General branching processes Multivariate stable laws Multitype branching random walk Choquet–Deny lemma Weighted branching Enthalten in Probability theory and related fields Springer Berlin Heidelberg, 1986 164(2015), 1-2 vom: 21. Jan., Seite 401-458 (DE-627)129382779 (DE-600)165783-5 (DE-576)01476914X 0178-8051 nnns volume:164 year:2015 number:1-2 day:21 month:01 pages:401-458 https://doi.org/10.1007/s00440-015-0615-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 164 2015 1-2 21 01 401-458 |
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10.1007/s00440-015-0615-y doi (DE-627)OLC2054639710 (DE-He213)s00440-015-0615-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Mentemeier, Sebastian verfasserin aut The fixed points of the multivariate smoothing transform 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. Smoothing transform Markov random walks General branching processes Multivariate stable laws Multitype branching random walk Choquet–Deny lemma Weighted branching Enthalten in Probability theory and related fields Springer Berlin Heidelberg, 1986 164(2015), 1-2 vom: 21. Jan., Seite 401-458 (DE-627)129382779 (DE-600)165783-5 (DE-576)01476914X 0178-8051 nnns volume:164 year:2015 number:1-2 day:21 month:01 pages:401-458 https://doi.org/10.1007/s00440-015-0615-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 164 2015 1-2 21 01 401-458 |
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10.1007/s00440-015-0615-y doi (DE-627)OLC2054639710 (DE-He213)s00440-015-0615-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Mentemeier, Sebastian verfasserin aut The fixed points of the multivariate smoothing transform 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. Smoothing transform Markov random walks General branching processes Multivariate stable laws Multitype branching random walk Choquet–Deny lemma Weighted branching Enthalten in Probability theory and related fields Springer Berlin Heidelberg, 1986 164(2015), 1-2 vom: 21. Jan., Seite 401-458 (DE-627)129382779 (DE-600)165783-5 (DE-576)01476914X 0178-8051 nnns volume:164 year:2015 number:1-2 day:21 month:01 pages:401-458 https://doi.org/10.1007/s00440-015-0615-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 164 2015 1-2 21 01 401-458 |
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10.1007/s00440-015-0615-y doi (DE-627)OLC2054639710 (DE-He213)s00440-015-0615-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Mentemeier, Sebastian verfasserin aut The fixed points of the multivariate smoothing transform 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. Smoothing transform Markov random walks General branching processes Multivariate stable laws Multitype branching random walk Choquet–Deny lemma Weighted branching Enthalten in Probability theory and related fields Springer Berlin Heidelberg, 1986 164(2015), 1-2 vom: 21. Jan., Seite 401-458 (DE-627)129382779 (DE-600)165783-5 (DE-576)01476914X 0178-8051 nnns volume:164 year:2015 number:1-2 day:21 month:01 pages:401-458 https://doi.org/10.1007/s00440-015-0615-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 164 2015 1-2 21 01 401-458 |
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510 VZ 17,1 ssgn The fixed points of the multivariate smoothing transform Smoothing transform Markov random walks General branching processes Multivariate stable laws Multitype branching random walk Choquet–Deny lemma Weighted branching |
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The fixed points of the multivariate smoothing transform |
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The fixed points of the multivariate smoothing transform |
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Mentemeier, Sebastian |
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Probability theory and related fields |
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Mentemeier, Sebastian |
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the fixed points of the multivariate smoothing transform |
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The fixed points of the multivariate smoothing transform |
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Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. © Springer-Verlag Berlin Heidelberg 2015 |
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Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. © Springer-Verlag Berlin Heidelberg 2015 |
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Abstract Let $$(\mathbf {T}_1, \mathbf {T}_2, \ldots )$$ be a sequence of random $$d\times d$$ matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors $$X$$ with nonnegative entries, satisfying *X=L∑i≥1TiXi+Q,$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$where $$\mathop {=}\limits ^{{\mathcal {L}}}$$ denotes equality of the corresponding laws, $$(X_i)_{i \ge 1}$$ are i.i.d. copies of $$X$$ and independent of $$(Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )$$. For $$d=1$$, this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions $$X$$ to (*) in the non-critical case, and existence results in the critical case. © Springer-Verlag Berlin Heidelberg 2015 |
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The fixed points of the multivariate smoothing transform |
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