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improved wording to explicitly state equivalence of measure
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Alex Simpson
Alex Simpson
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In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability law for $(X,Y)$, and $P_X \otimes P_Y$ is the product-measure of the probability laws for $X$ and $Y$.

Weaker notions of independence can be obtained by relaxing the equality on the right-hand side of (1) to equivalence of measure or absolute continuity of measure. This leads to three natural weakenings of independence, obtained by replacing the right-hand equality in (1) by, respectively: $$(2)~~P_{(X,Y)} \equiv P_X \otimes P_Y \enspace ,\quad (3)~~P_{(X,Y)} \ll P_X \otimes P_Y \enspace , \quad (4)~~P_{(X,Y)} \gg P_X \otimes P_Y \enspace .$$ Such weakenings are plausible candidates for contexts in which a qualitative notion of independence is required.

All three weakenings above appear in the literature. E.g., Hegland and Pestov (Additive models in high dimensions, ANZIAM J. 46, 2004/05) consider (2), under the name quasi-independence; Sudakov considers (3) (in the context of measurable decompositions), again under the name quasi-independence; and Hoffmann-Jørgensen et. al. consider (4) under the name weak independence (where this is contrasted with yet another notion of quasi-independence).

My question is whether, other than appearing as incidental concepts in references such as the above, there exist any systematic treatments in the literature of any or all of these weakened notions of independence. E.g., are there references establishing basic properties, discussing alternative formulations, etc.?

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability law for $(X,Y)$, and $P_X \otimes P_Y$ is the product-measure of the probability laws for $X$ and $Y$.

Weaker notions of independence can be obtained by relaxing the equality on the right-hand side of (1) to equivalence or absolute continuity of measure. This leads to three natural weakenings of independence, obtained by replacing the right-hand equality in (1) by, respectively: $$(2)~~P_{(X,Y)} \equiv P_X \otimes P_Y \enspace ,\quad (3)~~P_{(X,Y)} \ll P_X \otimes P_Y \enspace , \quad (4)~~P_{(X,Y)} \gg P_X \otimes P_Y \enspace .$$ Such weakenings are plausible candidates for contexts in which a qualitative notion of independence is required.

All three weakenings above appear in the literature. E.g., Hegland and Pestov (Additive models in high dimensions, ANZIAM J. 46, 2004/05) consider (2), under the name quasi-independence; Sudakov considers (3) (in the context of measurable decompositions), again under the name quasi-independence; and Hoffmann-Jørgensen et. al. consider (4) under the name weak independence (where this is contrasted with yet another notion of quasi-independence).

My question is whether, other than appearing as incidental concepts in references such as the above, there exist any systematic treatments in the literature of any or all of these weakened notions of independence. E.g., are there references establishing basic properties, discussing alternative formulations, etc.?

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability law for $(X,Y)$, and $P_X \otimes P_Y$ is the product-measure of the probability laws for $X$ and $Y$.

Weaker notions of independence can be obtained by relaxing the equality on the right-hand side of (1) to equivalence of measure or absolute continuity. This leads to three natural weakenings of independence, obtained by replacing the right-hand equality in (1) by, respectively: $$(2)~~P_{(X,Y)} \equiv P_X \otimes P_Y \enspace ,\quad (3)~~P_{(X,Y)} \ll P_X \otimes P_Y \enspace , \quad (4)~~P_{(X,Y)} \gg P_X \otimes P_Y \enspace .$$ Such weakenings are plausible candidates for contexts in which a qualitative notion of independence is required.

All three weakenings above appear in the literature. E.g., Hegland and Pestov (Additive models in high dimensions, ANZIAM J. 46, 2004/05) consider (2), under the name quasi-independence; Sudakov considers (3) (in the context of measurable decompositions), again under the name quasi-independence; and Hoffmann-Jørgensen et. al. consider (4) under the name weak independence (where this is contrasted with yet another notion of quasi-independence).

My question is whether, other than appearing as incidental concepts in references such as the above, there exist any systematic treatments in the literature of any or all of these weakened notions of independence. E.g., are there references establishing basic properties, discussing alternative formulations, etc.?

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Alex Simpson
Alex Simpson
  • 866
  • 4
  • 15

Qualitative weakenings of probabilistic independence

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability law for $(X,Y)$, and $P_X \otimes P_Y$ is the product-measure of the probability laws for $X$ and $Y$.

Weaker notions of independence can be obtained by relaxing the equality on the right-hand side of (1) to equivalence or absolute continuity of measure. This leads to three natural weakenings of independence, obtained by replacing the right-hand equality in (1) by, respectively: $$(2)~~P_{(X,Y)} \equiv P_X \otimes P_Y \enspace ,\quad (3)~~P_{(X,Y)} \ll P_X \otimes P_Y \enspace , \quad (4)~~P_{(X,Y)} \gg P_X \otimes P_Y \enspace .$$ Such weakenings are plausible candidates for contexts in which a qualitative notion of independence is required.

All three weakenings above appear in the literature. E.g., Hegland and Pestov (Additive models in high dimensions, ANZIAM J. 46, 2004/05) consider (2), under the name quasi-independence; Sudakov considers (3) (in the context of measurable decompositions), again under the name quasi-independence; and Hoffmann-Jørgensen et. al. consider (4) under the name weak independence (where this is contrasted with yet another notion of quasi-independence).

My question is whether, other than appearing as incidental concepts in references such as the above, there exist any systematic treatments in the literature of any or all of these weakened notions of independence. E.g., are there references establishing basic properties, discussing alternative formulations, etc.?