{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T17:45:25Z","timestamp":1785433525214,"version":"3.56.0"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T00:00:00Z","timestamp":1771545600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We investigate the limiting spectral distribution of a noncentral unified matrix model defined by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper Omega left parenthesis bold upper X right parenthesis equals left parenthesis left parenthesis bold upper X bold upper P 1 plus bold upper A right parenthesis left parenthesis bold upper X bold upper P 1 plus bold upper A right parenthesis Superscript prime Baseline divided by n 1 right parenthesis left parenthesis bold upper X bold upper P 2 bold upper X Superscript prime Baseline divided by n 2 right parenthesis Superscript negative 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi mathvariant=\"bold\">\u03a9<\/mml:mi>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">A<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">A<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>\u2212<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:math>\n                        <jats:tex-math>$\\boldsymbol{\\Omega}(\\mathbf{X}) = ({(\\mathbf{X}\\mathbf{P}_1+\\mathbf{A})(\\mathbf{X}\\mathbf{P}_1+\\mathbf{A})'}\/{n_1}) ({\\mathbf{X}\\mathbf{P}_2\\mathbf{X}'}\/{n_2})^{-1}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper X equals left parenthesis upper X Subscript i j Baseline right parenthesis Subscript p times n<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>i<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mo>\u00d7<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{X}=(X_{ij})_{p\\times n}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a random matrix with independent and identically distributed real entries having zero mean and finite second moment.\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper A<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold\">A<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{A}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline4.png\">\n                          <jats:alt-text content-type=\"machine-generated\">p times n<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>p<\/mml:mi>\n                          <mml:mo>\u00d7<\/mml:mo>\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$p\\times n$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    nonrandom matrix. The matrices\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline5.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper P 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{P}_1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline6.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper P 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{P}_2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    are projection matrices satisfying\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline7.png\">\n                          <jats:alt-text content-type=\"machine-generated\">normal r normal a normal n normal k left parenthesis bold upper P 1 right parenthesis equals n 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">r<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">a<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">n<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">k<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathrm{rank}(\\mathbf{P}_1)=n_1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline8.png\">\n                          <jats:alt-text content-type=\"machine-generated\">normal r normal a normal n normal k left parenthesis bold upper P 2 right parenthesis equals n 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">r<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">a<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">n<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">k<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathrm{rank}(\\mathbf{P}_2)=n_2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline9.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper P 1 bold upper P 2 equals 0<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:mn>0<\/mml:mn>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{P}_1\\mathbf{P}_2=0$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . When\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline10.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper P 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{P}_1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline11.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper P 2<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">P<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{P}_2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    are random, they are assumed to be independent of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline12.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper X<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold\">X<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                        <jats:tex-math>$\\mathbf{X}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . When\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline13.png\">\n                          <jats:alt-text content-type=\"machine-generated\">p divided by n 1 right arrow c 1 element of left parenthesis 0 comma normal infinity right parenthesis<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>p<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo stretchy=\"false\">\u2192<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>\u2208<\/mml:mo>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mn>0<\/mml:mn>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mi mathvariant=\"normal\">\u221e<\/mml:mi>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                        <\/mml:math>\n                        <jats:tex-math>$p\/n_1\\to c_1\\in(0,\\infty)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline14.png\">\n                          <jats:alt-text content-type=\"machine-generated\">p divided by n 2 right arrow c 2 element of left parenthesis 0 comma 1 right parenthesis<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi>p<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo stretchy=\"false\">\u2192<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>\u2208<\/mml:mo>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mn>0<\/mml:mn>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                        <\/mml:math>\n                        <jats:tex-math>$p\/n_2\\to c_2\\in(0,1)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , we establish the almost sure convergence of the empirical spectral distribution of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100570_inline15.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold upper Omega<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mi mathvariant=\"bold\">\u03a9<\/mml:mi>\n                        <\/mml:math>\n                        <jats:tex-math>$\\boldsymbol{\\Omega}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    to a deterministic limiting distribution. Furthermore, we show that this limiting distribution coincides with that of the noncentral F-matrix, thus revealing a deep connection between the proposed model and classical multivariate analysis.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10057","type":"journal-article","created":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T08:39:08Z","timestamp":1771576748000},"page":"772-787","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["The limiting spectral distribution of the noncentral unified matrix model"],"prefix":"10.1017","volume":"63","author":[{"given":"Zhining","family":"Wang","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02rkvz144","id-type":"ROR","asserted-by":"publisher"}],"name":"Northeast Normal University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuling","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/052pakb34","id-type":"ROR","asserted-by":"publisher"}],"name":"Changchun University of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiang","family":"Hu","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02rkvz144","id-type":"ROR","asserted-by":"publisher"}],"name":"Northeast Normal University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,2,20]]},"reference":[{"key":"S0021900225100570_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s10463-015-0514-0"},{"key":"S0021900225100570_ref4","doi-asserted-by":"publisher","DOI":"10.3150\/17-BEJ965"},{"key":"S0021900225100570_ref7","doi-asserted-by":"publisher","DOI":"10.1070\/SM1967v001n04ABEH001994"},{"key":"S0021900225100570_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v17-2239"},{"key":"S0021900225100570_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-0661-8"},{"key":"S0021900225100570_ref10","doi-asserted-by":"publisher","DOI":"10.1214\/14-AOS1284"},{"key":"S0021900225100570_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-020-1958-1"},{"key":"S0021900225100570_ref2","first-page":"612","article-title":"Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case","volume":"47","author":"Bao","year":"2015","journal-title":"Ann. Statist."},{"key":"S0021900225100570_ref8","first-page":"325","article-title":"On the distribution of the roots of certain symmetric matrices","volume":"67","author":"Wigner","year":"1958","journal-title":"Ann. Statist."},{"key":"S0021900225100570_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176997023"},{"key":"S0021900225100570_ref6","doi-asserted-by":"publisher","DOI":"10.1142\/S2010326324500254"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100570","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T08:36:32Z","timestamp":1782290192000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100570\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,20]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["S0021900225100570"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10057","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,20]]},"assertion":[{"value":"\u00a9 The Author(s), 2026. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}