Notions of Local Dependence Persistence ( k + 1) -block factor vs. k -dependence Proof of the lower bound Local Dependence and Persistence in Discrete Sliding Window Processes Ohad N. Feldheim Joint work with Noga Alon Weizmann Institute of Science July 2014 Technishe Universitat Darmstadt Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Discrete Sliding Window Processes Persistence Applications of Sliding Window Processes ( k + 1) -block factor vs. k -dependence k -dependence Proof of the lower bound Sliding Window Processes { Z t } t ∈ Z := i.i.d. uniform on [0 , 1]. f : [0 , 1] k → { 0 , . . . , r − 1 } measurable. { X t } t ∈ Z := f ( Z t , Z t +1 , . . . , Z t + k − 1 ) . Z t f f X t Such a process is called k -block factor . If r = 2 we call it a binary k -block factor. Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Discrete Sliding Window Processes Persistence Applications of Sliding Window Processes ( k + 1) -block factor vs. k -dependence k -dependence Proof of the lower bound Applications Sliding window processes have many real-life applications, e.g., Linguistics, Vocoding: • Model for voiceless phonemes Cryptography: • Encryption schemes with parallel decryption Computer science: • Data processes by stateless machines • Distributive ring computation Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Discrete Sliding Window Processes Persistence Applications of Sliding Window Processes ( k + 1) -block factor vs. k -dependence k -dependence Proof of the lower bound Local dependence k -dependence for stationary processes If every E − which is { X t } t < 0 measurable, and every E + which is { X t } t ≥ k measurable are independent, then { X t } is said to be k -dependent . Observation k + 1-block factors are stationary k -dependent. ) ) Z t ) ) X t Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Discrete Sliding Window Processes Persistence Applications of Sliding Window Processes ( k + 1) -block factor vs. k -dependence k -dependence Proof of the lower bound Local dependence k -dependence for stationary processes If every E − which is { X t } t < 0 measurable, and every E + which is { X t } t ≥ k measurable are independent, then { X t } is said to be k -dependent . Observation k + 1-block factors are stationary k -dependent. ) ) Z t ) ) X t Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Discrete Sliding Window Processes Persistence Applications of Sliding Window Processes ( k + 1) -block factor vs. k -dependence k -dependence Proof of the lower bound Local dependence k -dependence for stationary processes If every E − which is { X t } t < 0 measurable, and every E + which is { X t } t ≥ k measurable are independent, then { X t } is said to be k -dependent . Observation k + 1-block factors are stationary k -dependent. ) ) Z t ) ) X t Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Some previous results on block factors 2 -block factors Katz, 1971 Computed max P ( X 1 = X 2 = 1) given P ( X 1 = 1). De Valk, 1988 Computed min P ( X 1 = X 2 = 1) given P ( X 1 = 1) and showed uniqueness of the minimal and maximal processes. He did this also for general 1-dependent processes. Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Some previous results on block factors 2 -block factors Katz, 1971 Computed max P ( X 1 = X 2 = 1) given P ( X 1 = 1). De Valk, 1988 Computed min P ( X 1 = X 2 = 1) given P ( X 1 = 1) and showed uniqueness of the minimal and maximal processes. He did this also for general 1-dependent processes. k -block factors Janson, 1984: Explored several examples of binary k -block factors with at least k − 1 zeroes between consecutive ones, and showed convergence of the gaps between consecutive ones for such processes. Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Persistence A natural definition of persistence in a frame of size q , for processes with discrete image: � � P X q = P X 1 = X 2 = · · · = X q Coincides with the usual definition of persistence, if f ( Z 1 , . . . , Z k ) = 1 I { g ( Z 1 , . . . , Z k ) > 0 } , for some function g . Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Persistence A natural definition of persistence in a frame of size q , for processes with discrete image: � � P X q = P X 1 = X 2 = · · · = X q Coincides with the usual definition of persistence, if f ( Z 1 , . . . , Z k ) = 1 I { g ( Z 1 , . . . , Z k ) > 0 } , for some function g . Observation X is non-constant k -dependent → ∃ c > 0 s. t. P X q < e − cq Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Persistence A natural definition of persistence in a frame of size q , for processes with discrete image: � � P X q = P X 1 = X 2 = · · · = X q Coincides with the usual definition of persistence, if f ( Z 1 , . . . , Z k ) = 1 I { g ( Z 1 , . . . , Z k ) > 0 } , for some function g . Observation X is non-constant k -dependent → ∃ c > 0 s. t. P X q < e − cq But what about a lower bound? Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Lower bound if Z t ∈ { 0 , . . ., ℓ − 1 } Observation If we had Z t ∈ { 0 , . . . , ℓ − 1 } it would imply ℓ − ( q + k − 1) < P X q . ) ) ) ) Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Somewhat unusual question Usually: low correlation �→ lower bound on persistence. Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Somewhat unusual question Usually: low correlation �→ lower bound on persistence. Lower bound on block-factor persistence ← → There is a universal constant p k , q such that every symmetric real sliding window process { X t } t ∈ Z with a given window size k must have: � � X 1 , . . . , X q ∈ [0 , ∞ ) P > p k , q Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Somewhat unusual question Usually: low correlation �→ lower bound on persistence. Lower bound on block-factor persistence ← → There is a universal constant p k , q such that every symmetric real sliding window process { X t } t ∈ Z with a given window size k must have: � � X 1 , . . . , X q ∈ [0 , ∞ ) P > p k , q There is a block factor with P q = 0 for some q ← → Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Somewhat unusual question Usually: low correlation �→ lower bound on persistence. Lower bound on block-factor persistence ← → There is a universal constant p k , q such that every symmetric real sliding window process { X t } t ∈ Z with a given window size k must have: � � X 1 , . . . , X q ∈ [0 , ∞ ) P > p k , q There is a block factor with P q = 0 for some q ← → Each of N players, standing in a row is assigned a random number uniform in [0 , 1]. By looking only on the numbers in their q neighborhood, using a symmetric algorithm, the players can divide themselves to consecutive pairs and triplets. Ohad N. Feldheim Persistence in Sliding Window Processes
Notions of Local Dependence Persistence Previous results ( k + 1) -block factor vs. k -dependence Persistence in block factors Proof of the lower bound Our results Let k , q ∈ N . For f : R k → { 0 , 1 } write X f t = f ( Z t , . . . , Z t + k − 1 ) where Z t are i.i.d, and write � � p min X f 1 = X f 2 = · · · = X f = inf f { P } q q Theorem (Alon, F.) 1 1 ( T k − 2 ( q 2 )) k + q − 1 < p min < 100 ) , q T k − 2 ( q where ... 2 x T ℓ ( x ) := 2 2 2 � �� � ℓ times Ohad N. Feldheim Persistence in Sliding Window Processes
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