The existence of data with different dependence structures motivates the development of models which can capture several types of dependence. In this paper we consider a stationary sequence of moving maxima vectors <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{{\mathbf {X}}_n=(X_{n1},\ldots ,X_{nd})\}_{n\ge 1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="bold">X</mi> <mi>n</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation> having innovations <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\mathbf {Z}}_{l,n}=(Z_{l,n,1},\ldots ,Z_{l,n,d})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi mathvariant="bold">Z</mi> <mrow> <mi>l</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mrow> <mi>l</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Z</mi> <mrow> <mi>l</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> with totally dependent margins for certain values of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$l,$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>l</mi> <mo>,</mo> </mrow> </math> </EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$l\in I_1,$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>l</mi> <mo>∈</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </math> </EquationSource> </InlineEquation> and independent margins for the remaining values of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$l,$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>l</mi> <mo>,</mo> </mrow> </math> </EquationSource> </InlineEquation> <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$l\in I_2.$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>l</mi> <mo>∈</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>.</mo> </mrow> </math> </EquationSource> </InlineEquation> We obtain in this way a <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$d$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>d</mi> </math> </EquationSource> </InlineEquation>-dimensional process <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\{{\mathbf {X}}_n\}_{n\ge 1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="bold">X</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation> whose extremal dependence, measured by the tail dependence coefficients, lies between asymptotic independence and total dependence. The extremal properties of these M4 processes are studied and examined both theoretically and through simulation studies: we derive the multivariate extremal index, the tail dependence coefficients and co-movement indices. Copyright Sociedad de Estadística e Investigación Operativa 2014