# Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k

Assume $$X_1,...,X_t$$ are independent distributions on $$\mathbb{Z}_2^n$$ s.t. each $$X_i$$ is $$\epsilon$$-close to min-entropy $$k$$; i.e. there exist distributions $$Y_1,...,Y_t$$ on $$\mathbb{Z}_2^n$$ s.t: $$\forall i: \frac{1}{2}\sum_{x\in\mathbb{Z}_2^n}|X_i(x)-Y_i(x)|\leq\epsilon$$ and: $$\forall i: H_\infty(Y_i)\geq k$$ where $$H_\infty(Y_i)=\min_{x\in\mathbb{Z}_2^n}\{-\log_2(Y_i(x))\}$$ is the min-entropy of $$Y_i$$.

Define $$X=\sum_{i=1}^t X_i$$ as the distribution corresponding to the convolution of $$X_1,...,X_t$$: $$X(x)=\sum_{\substack{x_1,...,x_t\in\mathbb{Z}_2^n\\x_1\oplus...\oplus x_t=x}}\prod_{i=1}^t X_i(x_i)$$ How do I show that $$X$$ is $$\epsilon^t$$-close to min-entropy $$k$$?