The Hewitt–Savage zero–one law is a theorem in probability theory, similar to Kolmogorov's zero–one law and the Borel–Cantelli lemma, that specifies that a certain type of event will either almost surely happen or almost surely not happen. It is sometimes known as the Savage-Hewitt law for symmetric events. It is named after Edwin Hewitt and Leonard Jimmie Savage.
Statement of the Hewitt-Savage zero-one law Let { X n } n = 1 ∞ {\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }} be a sequence of independent and identically distributed random variables taking values in a set X {\displaystyle \mathbb {X} } . The Hewitt-Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite permutations of the indices, has probability either 0 or 1 (a “finite” permutation is one that leaves all but finitely many of the indices fixed). Somewhat more abstractly, define the exchangeable sigma algebra or sigma algebra of symmetric events E {\displaystyle {\mathcal {E}}} to be the set of events (depending on the sequence of variables { X n } n = 1 ∞ {\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }} ) which are invariant under finite permutations of the indices in the sequence { X n } n = 1 ∞ {\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }} . Then A ∈ E ⟹ P ( A ) ∈ { 0 , 1 } {\displaystyle A\in {\mathcal {E}}\implies \mathbb {P} (A)\in \{0,1\}} . Since any finite permutation can be written as a product of transpositions, if we wish to check whether or not an event A {\displaystyle A} is symmetric (lies in E {\displaystyle {\mathcal {E}}} ), it is enough to check if its occurrence is unchanged by an arbitrary transposition ( i , j ) {\displaystyle (i,j)} , i , j ∈ N {\displaystyle i,j\in \mathbb {N} } .
Example Let the sequence { X n } n = 1 ∞ {\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }} of independent and identically distributed random variables taking values in R {\displaystyle \mathbb {R} } . Consider the random walk S N = ∑ n = 1 N X n {\displaystyle S_{N}=\sum _{n=1}^{N}X_{n}} . Then one of the following occurs with probability 1:
S N = 0 {\displaystyle S_{N}=0}
S N → ∞ {\displaystyle S_{N}\to \infty }
S N → − ∞ {\displaystyle S_{N}\to -\infty }
lim inf S N = − ∞ {\displaystyle \liminf S_{N}=-\infty } and lim sup S N = ∞ {\displaystyle \limsup S_{N}=\infty } . Since SN are not independent the Kolmogorov's zero–one law is not directly applicable. First consider the case when X1 is a.s. constant. Then with probability 1 we have that either ( S N = 0 {\displaystyle S_{N}=0} ), ( S N → ∞ {\displaystyle S_{N}\to \infty } ) or ( S N → − ∞ {\displaystyle S_{N}\to -\infty } ). Now consider the case, when X1 is not a.s. constant. Then for any t ∈ [ − ∞ , ∞ ] {\displaystyle t\in [-\infty ,\infty ]} the event { lim sup S N ≥ t } {\displaystyle \left\{\limsup S_{N}\geq t\right\}} is in the exchangeable sigma algebra. That is because limit supremum does not change with finite permutation of the indices. From Hewitt-Savage zero-one law we have that
P ( lim sup S N ≥ t ) ∈ { 0 , 1 } {\displaystyle \mathbb {P} \left(\limsup S_{N}\geq t\right)\in \{0,1\}} . There has to exist t, where probability switches from 0 to 1 i.e. exists t ∗ ∈ [ − ∞ , ∞ ] {\displaystyle t^{*}\in [-\infty ,\infty ]} such that lim sup S n = t ∗ {\displaystyle \limsup S_{n}=t^{*}} almost surely. Similarly exists t ∗ ∈ [ − ∞ , ∞ ] {\displaystyle t_{*}\in [-\infty ,\infty ]} such that lim inf S n = t ∗ {\displaystyle \liminf S_{n}=t_{*}} almost surely. Since almost surely
t ∗ = lim sup S N = X 1 + lim sup ∑ n = 2 N X n = X 1 + t ∗ {\displaystyle t^{*}=\limsup S_{N}=X_{1}+\limsup \sum _{n=2}^{N}X_{n}=X_{1}+t^{*}}
and X1 is not a.s. 0, then t ∗ {\displaystyle t^{*}} is not finite. Similarly t ∗ {\displaystyle t_{*}} in not finite. Therefore, with probability 1 either ( S N → ∞ {\displaystyle S_{N}\to \infty } ), ( S N → − ∞ {\displaystyle S_{N}\to -\infty } ) or ( lim inf S N = − ∞ {\displaystyle \liminf S_{N}=-\infty } and lim sup S N = ∞ {\displaystyle \limsup S_{N}=\infty } ).
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