The results can be used to derive limit theorems with rates of convergence for marked and thinned point processes. In particular, a new lower bound for the total variation distance between a Bernoulli partial sum process and the accompanying Poisson process is obtained. 2 d TV(P;Q) = jP Qj 1 def= X x2X jP(x) Q(x)j Proof. De nition 2. 4.2.1 Bounding the total variation distance via coupling Let µ and ⌫ be probability measures on (S,S). Lemma 4.9 (Coupling inequality). Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Some examples are given. The total variation distance between two probability measures P and Q on a sigma-algebra $${\displaystyle {\mathcal {F}}}$$ of subsets of the sample space $${\displaystyle \Omega }$$ is defined via Let µ and ⌫ be probability measures on (S,S). The total variation distance between P, and Qis d TV(P;Q) = sup A X jP(A) Q(A)j: The TV distance is related to the ‘ 1 distance as follows: Claim 3. The rate of closeness is studied in terms of the minimal distance in probability. pling is also useful to bound the distance between probability measures. 1.1 Total Variation/‘ 1 distance For a subset A X, let P(A) = P x2A P(x) be the probability of observing an element in A. Recall the definition of the total variation distance kµ⌫k TV:= sup A2S |µ(A)⌫(A)|. Abstract Bounds for the total variation distance between the distribution of the sum of a random number of Bernoulli summands and an appropriate Poisson distribution are given.